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w(462)p 24107 31324 V 1980 w(630)p 28037 31324 V 1980 w(420)p 31967 31324 V 1979 w(140)p 35897 31324 V 1980 w(21)p 39176 31324 V 1980 w(1)p 41805 31324 V 43880 31324 V 11889 32929 V 14042 32448 a Fe(7)p 16898 32929 V 2796 w Fu(429)p 20178 32929 V 1329 w(1716)p 24107 32929 V 1330 w(2772)p 28037 32929 V 1329 w(2310)p 31967 32929 V 1330 w(1050)p 35897 32929 V 1330 w(252)p 39176 32929 V 1329 w(28)p 41805 32929 V 1426 w(1)p 43880 32929 V 11911 32974 31992 45 v 15134 36164 a(T)-108 b(able)434 b(1:)579 b(Some)433 b(of)h(the)f(co)36 b(e\261cien)-36 b(ts)868 b Fj(e)p Fu(\()p Fj(m;)221 b(q)48 b Fu(\))2751 38995 y(Equation)434 b(\()p 0 1 0 0 TeXcolorcmyk(4)p (#equation.4) [[152 366 158 378] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))f(can)h(b)36 b(e)433 b(in)-36 b(tro)36 b(duced)433 b(in)-36 b(to)433 b(\()p 0 1 0 0 TeXcolorcmyk(1)p (#equation.1) [[290 366 296 378] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))h(to)f(obtain)5798 41283 y Fm(m)p Fn(+1)5856 41682 y Fh(X)5988 44472 y Fm(q)32 b Fn(=0)8054 42944 y Fj(e)p Fu(\()p Fj(m)294 b Fu(+)h(1)p Fj(;)221 b(q)48 b Fu(\))434 b Fj(y)15381 42395 y Fm(q)16257 42944 y Fu(=)18181 41283 y Fm(m)17637 41682 y Fh(X)17770 44472 y Fm(q)32 b Fn(=0)19778 42944 y Fj(e)p Fu(\()p Fj(m;)221 b(q)48 b Fu(\))432 b Fj(y)24852 42395 y Fm(q)32 b Fn(+1)26857 42944 y Fu(+)28707 41283 y Fm(m)28164 41682 y Fh(X)28304 44472 y Fm(s)p Fn(=0)30304 41283 y Fm(m)p Fg(\241)p Fm(s)30344 41682 y Fh(X)30465 44472 y Fm(p)p Fn(=0)32524 42944 y Fj(e)p Fu(\()p Fj(m)295 b Fd(\241)g Fj(s;)221 b(p)p Fu(\))40145 41283 y Fm(s)39403 41682 y Fh(X)39536 44472 y Fm(r)26 b Fn(=0)41543 42944 y Fj(e)p Fu(\()p Fj(s;)221 b(r)36 b Fu(\))434 b Fj(y)46091 42395 y Fm(p)p Fn(+)p Fm(r)800 46952 y Fu(and)f(transformed)g(as)h(follo)-36 b(ws:)1354 48425 y Fh(P)2756 48775 y Fm(m)p Fn(+1)2756 49809 y Fm(q)32 b Fn(=0)5067 49421 y Fj(e)p Fu(\()p Fj(m)295 b Fu(+)f(1)p Fj(;)221 b(q)48 b Fu(\))434 b Fj(y)12394 48939 y Fm(q)13270 49421 y Fu(=)14651 48425 y Fh(P)16053 48775 y Fm(m)p Fn(+1)16053 49809 y Fm(q)32 b Fn(=1)18364 49421 y Fj(e)p Fu(\()p Fj(m;)221 b(q)343 b Fd(\241)295 b Fu(1\))434 b Fj(y)25713 48939 y Fm(q)26515 49421 y Fu(+)27822 48425 y Fh(P)29224 48775 y Fm(m)29224 49809 y(s)p Fn(=0)31138 48425 y Fh(P)32540 48775 y Fm(m)p Fg(\241)p Fm(s)32540 49809 y(p)p Fn(=0)34816 49421 y Fj(e)p Fu(\()p Fj(m)295 b Fd(\241)g Fj(s;)221 b(p)p Fu(\))41695 48425 y Fh(P)43097 48775 y Fm(s)43097 49809 y(r)26 b Fn(=0)45027 49421 y Fj(e)p Fu(\()p Fj(s;)221 b(r)36 b Fu(\))434 b Fj(y)49575 48939 y Fm(p)p Fn(+)p Fm(r)51655 49421 y Fu(=)6094 51197 y(=)7474 50201 y Fh(P)8877 50551 y Fm(m)p Fn(+1)8877 51585 y Fm(q)32 b Fn(=1)11188 51197 y Fj(e)p Fu(\()p Fj(m;)221 b(q)343 b Fd(\241)295 b Fu(1\))434 b Fj(y)18537 50715 y Fm(q)19339 51197 y Fu(+)20646 50201 y Fh(P)22048 50551 y Fm(m)22048 51585 y(p)p Fn(=0)24434 50201 y Fh(P)25836 50551 y Fm(m)p Fg(\241)p Fm(p)25836 51585 y(r)26 b Fn(=0)28584 51197 y Fj(y)29266 50715 y Fm(p)p Fn(+)p Fm(r)31633 50121 y Fh(\243)32186 50201 y(P)33588 50551 y Fm(m)p Fg(\241)p Fm(p)33588 51585 y(s)p Fn(=)p Fm(r)35903 51197 y Fj(e)p Fu(\()p Fj(m)294 b Fd(\241)i Fj(s;)221 b(p)p Fu(\))433 b Fj(e)p Fu(\()p Fj(s;)221 b(r)36 b Fu(\))45992 50121 y Fh(\244)46915 51197 y Fu(=)5280 53284 y(=)6661 52288 y Fh(P)8063 52639 y Fm(m)p Fn(+1)8063 53672 y Fm(q)c Fn(=1)10374 53284 y Fj(e)p Fu(\()p Fj(m;)221 b(q)343 b Fd(\241)295 b Fu(1\))434 b Fj(y)17723 52802 y Fm(q)18525 53284 y Fu(+)19832 52288 y Fh(P)21234 52639 y Fm(m)21234 53672 y(p)p Fn(=0)23620 52288 y Fh(P)25022 52639 y Fm(m)25022 53672 y(q)32 b Fn(=)p Fm(p)27390 53284 y Fj(y)28072 52802 y Fm(q)29234 51810 y Fh(h)29861 52288 y(P)31263 52639 y Fm(m)p Fg(\241)p Fm(p)31263 53672 y(s)p Fn(=)p Fm(q)g Fg(\241)p Fm(p)34364 53284 y Fj(e)p Fu(\()p Fj(m)295 b Fd(\241)g Fj(s;)221 b(p)p Fu(\))434 b Fj(e)p Fu(\()p Fj(s;)221 b(q)342 b Fd(\241)296 b Fj(p)p Fu(\))46733 51810 y Fh(i)47728 53284 y Fu(=)5282 55676 y(=)6662 54679 y Fh(P)8064 55030 y Fm(m)p Fn(+1)8064 56063 y Fm(q)32 b Fn(=1)10376 55676 y Fj(e)p Fu(\()p Fj(m;)221 b(q)342 b Fd(\241)296 b Fu(1\))433 b Fj(y)17724 55193 y Fm(q)18527 55676 y Fu(+)19833 54679 y Fh(P)21235 55030 y Fm(m)21235 56063 y(q)32 b Fn(=0)23600 54679 y Fh(P)25002 55030 y Fm(q)25002 56063 y(p)p Fn(=0)27822 54201 y Fh(h)28449 54679 y(P)29851 55030 y Fm(m)p Fg(\241)p Fm(p)29851 56063 y(s)p Fn(=)p Fm(q)g Fg(\241)p Fm(p)32952 55676 y Fj(e)p Fu(\()p Fj(m)295 b Fd(\241)g Fj(s;)221 b(p)p Fu(\))434 b Fj(e)p Fu(\()p Fj(s;)221 b(q)342 b Fd(\241)296 b Fj(p)p Fu(\))45321 54201 y Fh(i)46169 55676 y Fj(y)46851 55193 y Fm(q)47727 55676 y Fu(=)4396 58067 y(=)5777 57070 y Fh(P)7179 57421 y Fm(m)p Fn(+1)7179 58454 y Fm(q)32 b Fn(=1)9490 58067 y Fj(e)p Fu(\()p Fj(m;)221 b(q)343 b Fd(\241)295 b Fu(1\))434 b Fj(y)16839 57585 y Fm(q)17641 58067 y Fu(+)18948 57070 y Fh(P)20350 57421 y Fm(m)20350 58454 y(q)32 b Fn(=0)22715 57070 y Fh(P)24117 57421 y Fm(q)24117 58454 y(p)p Fn(=0)26937 56592 y Fh(h)27564 57070 y(P)28966 57421 y Fm(m)28966 58454 y(l)11 b Fn(=)p Fm(q)30718 58067 y Fj(e)p Fu(\()p Fj(m)294 b Fd(\241)i Fj(l)325 b Fu(+)294 b Fj(p;)221 b(p)p Fu(\))434 b Fj(e)p Fu(\()p Fj(l)324 b Fd(\241)296 b Fj(p;)221 b(q)343 b Fd(\241)295 b Fj(p)p Fu(\))47225 56592 y Fh(i)48073 58067 y Fj(y)48755 57585 y Fm(q)49262 58067 y Fj(:)800 61105 y Fu(The)434 b(follo)-36 b(wing)435 b(set)f(of)g(equations)g(can)g (then)e(b)36 b(e)434 b(obtained)f(for)h(an)-36 b(y)434 b(natural)f(n)-36 b(um)g(b)36 b(er)432 b Fj(m)p Fu(:)2751 62710 y(\(1\))i(for)g Fj(q)416 b Fu(=)369 b(0:)16498 65328 y Fj(e)p Fu(\()p Fj(m)295 b Fu(+)g(1)p Fj(;)221 b Fu(0\))370 b(=)25029 63667 y Fm(m)24485 64066 y Fh(X)24626 66855 y Fm(s)p Fn(=0)26625 65328 y Fj(e)p Fu(\()p Fj(m)295 b Fd(\241)g Fj(s;)221 b Fu(0\))434 b Fj(e)p Fu(\()p Fj(s;)221 b Fu(0\))p Fj(;)800 68661 y Fu(whose)434 b(solution)g(is)18268 70698 y Fj(e)p Fu(\()p Fj(m;)221 b Fu(0\))370 b(=)e Fj(C)24934 70897 y Fm(m)26190 70698 y Fu(=)29074 69799 y(1)p 27704 70392 3391 54 v 27704 71609 a Fj(m)295 b Fu(+)g(1)31227 68825 y Fh(\265)32205 69799 y Fu(2)p Fj(m)32530 71609 y(m)33993 68825 y Fh(\266)34970 70698 y Fj(:)15807 b Fu(\(5\))p Black Black eop %%Page: 3 3 3 2 bop Black 0 TeXcolorgray Black 52150 -2672 a Fu(3)p Black 2751 1424 a(\(2\))434 b(for)g(0)369 b Fj(<)g(q)416 b Fd(\267)369 b Fj(m)p Fu(:)7894 5003 y Fj(e)p Fu(\()p Fj(m)295 b Fu(+)g(1)p Fj(;)221 b(q)48 b Fu(\))369 b(=)f Fj(e)p Fu(\()p Fj(m;)221 b(q)343 b Fd(\241)295 b Fu(1\))h(+)24424 3289 y Fm(q)23690 3742 y Fh(X)23812 6531 y Fm(p)p Fn(=0)27241 3343 y Fm(m)26698 3742 y Fh(X)26919 6570 y Fm(l)11 b Fn(=)p Fm(q)28838 5003 y Fj(e)p Fu(\()p Fj(m)294 b Fd(\241)i Fj(l)324 b Fu(+)295 b Fj(p;)221 b(p)p Fu(\))434 b Fj(e)p Fu(\()p Fj(l)324 b Fd(\241)296 b Fj(p;)221 b(q)343 b Fd(\241)295 b Fj(p)p Fu(\))p Fj(:)5432 b Fu(\(6\))2751 8890 y(\(3\))434 b(for)g Fj(q)416 b Fu(=)369 b Fj(m)295 b Fu(+)f(1:)16170 10495 y Fj(e)p Fu(\()p Fj(m)h Fu(+)f(1)p Fj(;)221 b(m)296 b Fu(+)f(1\))369 b(=)g Fj(e)p Fu(\()p Fj(m;)221 b(m)p Fu(\))368 b(=)h Fd(\242)221 b(\242)g(\242)370 b Fu(=)e(1)p Fj(:)13709 b Fu(\(7\))2751 12640 y(It)434 b(is)g(useful)f(to)h(in)-36 b(tro)36 b(duce)433 b(the)g(mo)36 b(di\257ed)433 b(matrix)h Fj(g)48 b Fu(\()p Fj(n;)221 b(k)45 b Fu(\))434 b(de\257ned)e(as)i(follo)-36 b(ws:)21035 15114 y Fj(g)48 b Fu(\()p Fj(n;)221 b(k)45 b Fu(\))370 b(=)e Fj(e)p Fu(\()p Fj(n)295 b Fu(+)g Fj(k)45 b(;)221 b(k)45 b Fu(\))21025 16719 y Fj(e)p Fu(\()p Fj(n;)221 b(k)45 b Fu(\))369 b(=)g Fj(g)48 b Fu(\()p Fj(n)295 b Fd(\241)g Fj(k)45 b(;)221 b(k)45 b Fu(\))51138 15928 y(\(8\))800 19142 y(T)-108 b(able)434 b(2)g(rep)36 b(orts)433 b(the)g(\257rst)g(few)h(v)-72 b(alues:)p Black Black 4481 22768 46852 45 v 4459 24373 45 1606 v 5145 23892 a Fj(m)369 b Ff(\302)g Fj(q)p 9468 24373 V 2579 w Fe(0)p 12747 24373 V 3183 w(1)p 16677 24373 V 3833 w(2)p 21257 24373 V 4483 w(3)p 26487 24373 V 5134 w(4)p 32368 24373 V 5134 w(5)p 38249 24373 V 5784 w(6)p 44780 24373 V 5784 w(7)p 51311 24373 V 4481 24418 46852 45 v 4459 26023 45 1606 v 6612 25541 a(0)p 9468 26023 V 4096 w Fu(1)p 12747 26023 V 3280 w(1)p 16677 26023 V 3930 w(1)p 21257 26023 V 4580 w(1)p 26487 26023 V 5231 w(1)p 32368 26023 V 5230 w(1)p 38249 26023 V 5881 w(1)p 44780 26023 V 5881 w(1)p 51311 26023 V 4459 27628 V 6612 27146 a Fe(1)p 9468 27628 V 4096 w Fu(1)p 12747 27628 V 3280 w(3)p 16677 27628 V 3930 w(6)p 21257 27628 V 3930 w(10)p 26487 27628 V 4580 w(15)p 32368 27628 V 4581 w(21)p 38249 27628 V 5231 w(28)p 44780 27628 V 5231 w(36)p 51311 27628 V 4459 29233 V 6612 28751 a Fe(2)p 9468 29233 V 4096 w Fu(2)p 12747 29233 V 2629 w(10)p 16677 29233 V 3280 w(30)p 21257 29233 V 3931 w(70)p 26487 29233 V 3930 w(140)p 32368 29233 V 3931 w(252)p 38249 29233 V 4581 w(420)p 44780 29233 V 4581 w(660)p 51311 29233 V 4459 30838 V 6612 30356 a Fe(3)p 9468 30838 V 4096 w Fu(5)p 12747 30838 V 2629 w(35)p 16677 30838 V 2630 w(140)p 21257 30838 V 3280 w(420)p 26487 30838 V 3281 w(1050)p 32368 30838 V 3280 w(2310)p 38249 30838 V 3931 w(4620)p 44780 30838 V 3931 w(8580)p 51311 30838 V 4459 32443 V 6612 31961 a Fe(4)p 9468 32443 V 3446 w Fu(14)p 12747 32443 V 1979 w(126)p 16677 32443 V 2630 w(630)p 21257 32443 V 2630 w(2310)p 26487 32443 V 3281 w(6930)p 32368 32443 V 2630 w(18018)p 38249 32443 V 3281 w(42042)p 44780 32443 V 3281 w(90090)p 51311 32443 V 4459 34048 V 6612 33567 a Fe(5)p 9468 34048 V 3446 w Fu(42)p 12747 34048 V 1979 w(462)p 16677 34048 V 1980 w(2772)p 21257 34048 V 1980 w(12012)p 26487 34048 V 2630 w(42042)p 32368 34048 V 1981 w(126126)p 38249 34048 V 2631 w(336336)p 44780 34048 V 2631 w(816816)p 51311 34048 V 4459 35653 V 6612 35172 a Fe(6)p 9468 35653 V 2795 w Fu(132)p 12747 35653 V 1330 w(1716)p 16677 35653 V 1329 w(12012)p 21257 35653 V 1981 w(60060)p 26487 35653 V 1980 w(240240)p 32368 35653 V 1981 w(816816)p 38249 35653 V 1980 w(2450448)p 44780 35653 V 1981 w(6651216)p 51311 35653 V 4459 37258 V 6612 36777 a Fe(7)p 9468 37258 V 2795 w Fu(429)p 12747 37258 V 1330 w(6435)p 16677 37258 V 1329 w(51480)p 21257 37258 V 1330 w(291720)p 26487 37258 V 1331 w(1312740)p 32368 37258 V 1330 w(4988412)p 38249 37258 V 1331 w(16628040)p 44780 37258 V 1331 w(49884120)p 51311 37258 V 4481 37303 46852 45 v 12947 39317 a(T)-108 b(able)434 b(2:)1012 b(Some)433 b(v)-72 b(alues)435 b(of)f(the)f(co)36 b(e\261cien)-36 b(ts)867 b Fj(g)48 b Fu(\()p Fj(m;)221 b(q)48 b Fu(\))800 41881 y(Equations)434 b(\()p 0 1 0 0 TeXcolorcmyk(5)p (#equation.5) [[139 340 145 352] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\),)g(\()p 0 1 0 0 TeXcolorcmyk(6)p (#equation.6) [[161 340 167 352] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\),)g(\()p 0 1 0 0 TeXcolorcmyk(7)p (#equation.7) [[183 340 189 352] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))f(then)g(b)36 b(ecome:)2751 43486 y(\(1\))434 b(for)g Fj(q)416 b Fu(=)369 b(0:)23120 45091 y Fj(g)48 b Fu(\()p Fj(n;)221 b Fu(0\))370 b(=)f Fj(C)29493 45290 y Fm(n)30118 45091 y Fj(:)20659 b Fu(\(9\))2751 47236 y(\(2\))434 b(for)g(0)369 b Fj(<)g(q)416 b Fd(\267)369 b Fj(n)296 b Fu(+)e(1:)9463 50815 y Fj(g)48 b Fu(\()p Fj(n)295 b Fu(+)g(1)p Fj(;)221 b(q)48 b Fu(\))369 b(=)g Fj(g)48 b Fu(\()p Fj(n)295 b Fu(+)f(1)p Fj(;)221 b(q)344 b Fd(\241)295 b Fu(1\))h(+)27658 49100 y Fm(q)26924 49553 y Fh(X)27046 52343 y Fm(p)p Fn(=0)30606 49154 y Fm(n)29932 49553 y Fh(X)30144 52382 y Fm(l)11 b Fn(=0)32072 50815 y Fj(g)48 b Fu(\()p Fj(n)295 b Fd(\241)g Fj(l)29 b(;)221 b(p)p Fu(\))435 b Fj(q)48 b Fu(\()p Fj(l)29 b(;)221 b(q)343 b Fd(\241)295 b Fj(p)p Fu(\))p Fj(;)6351 b Fu(\(10\))800 54662 y(where)434 b Fj(m)294 b Fd(\241)i Fj(q)481 b Fu(w)-36 b(as)434 b(replaced)f(b)-36 b(y)434 b Fj(n)369 b Fu(=)g Fj(m)295 b Fd(\241)g Fj(q)48 b Fu(.)578 b(Moreo)-36 b(v)g(er,)434 b(from)g(\()p 0 1 0 0 TeXcolorcmyk(7)p (#equation.7) [[381 225 387 237] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))f(w)-36 b(e)434 b(get)g Fj(g)48 b Fu(\(0)p Fj(;)221 b(n)p Fu(\))370 b(=)e(1)p Fj(:)2751 56267 y Fu(The)434 b(solution)f(of)i(\()p 0 1 0 0 TeXcolorcmyk(10)p (#equation.10) [[183 211 195 223] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))f(can)f(b)36 b(e)434 b(written)f(as)h(follo)-36 b(ws:)16059 59544 y Fj(g)48 b Fu(\()p Fj(n;)221 b(q)48 b Fu(\))368 b(=)22356 58646 y(\(2)p Fj(n)296 b Fu(+)f Fj(q)48 b Fu(\))o(!)p 21607 59239 6525 54 v 21607 60455 a Fj(q)g Fu(!)p Fj(n)p Fu(!\()p Fj(n)296 b Fu(+)e(1\)!)28634 59544 y(=)368 b Fj(C)30945 59743 y Fm(n)31571 57671 y Fh(\265)32549 58646 y Fu(2)p Fj(n)296 b Fu(+)f Fj(q)34063 60455 y(q)36202 57671 y Fh(\266)37180 59544 y Fj(:)12947 b Fu(\(11\))800 62821 y(In)433 b(fact,)i(\()p 0 1 0 0 TeXcolorcmyk(9)p (#equation.9) [[125 152 131 164] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))e(is)h(satis\257ed,)g(and)f(substituting)f(\()p 0 1 0 0 TeXcolorcmyk(11)p (#equation.11) [[292 152 303 164] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))i(in)-36 b(to)434 b(\()p 0 1 0 0 TeXcolorcmyk(10)p (#equation.10) [[340 152 352 164] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\),)g(w)-36 b(e)434 b(get)5260 66098 y Fj(C)6191 66297 y Fm(n)p Fn(+1)8019 64225 y Fh(\265)8997 65199 y Fu(2\()p Fj(n)296 b Fu(+)e(1\))i(+)e Fj(q)12143 67009 y(q)15914 64225 y Fh(\266)17999 66098 y Fu(=)1106 b Fj(C)21048 66297 y Fm(n)p Fn(+1)22876 64225 y Fh(\265)23854 65199 y Fu(2\()p Fj(n)296 b Fu(+)f(1\))g(+)g Fj(q)342 b Fd(\241)296 b Fu(1)27000 67009 y Fj(q)343 b Fd(\241)295 b Fu(1)33045 64225 y Fh(\266)34318 66098 y Fu(+)p Black 15158 w(\(12\))p Black 20117 70127 a(+)22024 68467 y Fm(n)21350 68865 y Fh(X)21563 71694 y Fm(l)11 b Fn(=0)23490 70127 y Fj(C)24421 70326 y Fm(n)p Fg(\241)p Fm(l)26070 70127 y Fj(C)27001 70326 y Fm(l)27569 67856 y Fh(")29077 68413 y Fm(q)28344 68865 y Fh(X)28465 71655 y Fm(p)p Fn(=0)31351 68254 y Fh(\265)32329 69229 y Fu(2\()p Fj(n)296 b Fd(\241)f Fj(l)29 b Fu(\))296 b(+)e Fj(p)35369 71038 y(p)39062 68254 y Fh(\266\265)41018 69229 y Fu(2)p Fj(l)325 b Fu(+)295 b Fj(q)342 b Fd(\241)296 b Fj(p)42352 71038 y(q)343 b Fd(\241)295 b Fj(p)46588 68254 y Fh(\266)47565 67856 y(#)p Black Black eop %%Page: 4 4 4 3 bop Black 0 TeXcolorgray Black 52150 -2672 a Fu(4)p Black 2751 1424 a(It)434 b(is)g(p)36 b(ossible)433 b(to)h(sho)-36 b(w)434 b(that)f(\(see)h(app)36 b(endix\))12937 3709 y Fm(q)12203 4162 y Fh(X)12325 6952 y Fm(p)p Fn(=0)15211 3551 y Fh(\265)16188 4525 y Fu(2\()p Fj(n)296 b Fd(\241)f Fj(l)29 b Fu(\))296 b(+)f Fj(p)19229 6335 y(p)22922 3551 y Fh(\266\265)24877 4525 y Fu(2)p Fj(l)326 b Fu(+)294 b Fj(q)343 b Fd(\241)295 b Fj(p)26212 6335 y(q)342 b Fd(\241)296 b Fj(p)30447 3551 y Fh(\266)31794 5424 y Fu(=)33174 3551 y Fh(\265)34152 4525 y Fu(2)p Fj(n)g Fu(+)f Fj(q)343 b Fu(+)294 b(1)36793 6335 y Fj(q)40058 3551 y Fh(\266)41035 5424 y Fj(;)800 9618 y Fu(and)3991 11110 y Fh(\265)4968 12085 y Fu(2\()p Fj(n)i Fu(+)f(1\))g(+)g Fj(q)8115 13894 y(q)11885 11110 y Fh(\266)13158 12983 y Fd(\241)14487 11110 y Fh(\265)15465 12085 y Fu(2\()p Fj(n)g Fu(+)g(1\))g(+)g Fj(q)343 b Fd(\241)295 b Fu(1)18611 13894 y Fj(q)342 b Fd(\241)296 b Fu(1)24655 11110 y Fh(\266)26002 12983 y Fu(=)27383 11110 y Fh(\265)28361 12085 y Fu(2\()p Fj(n)f Fu(+)g(1\))h(+)e Fj(q)343 b Fd(\241)295 b Fu(1)31507 13894 y Fj(q)343 b Fd(\241)295 b Fu(1)37552 11110 y Fh(\266)221 b(\267)39585 12085 y Fu(2\()p Fj(n)295 b Fu(+)g(1\))h(+)e Fj(q)p 39585 12678 6918 54 v 42731 13894 a(q)46930 12983 y Fd(\241)h Fu(1)48908 11110 y Fh(\270)5924 17776 y Fu(=)7305 15903 y Fh(\275)8434 16878 y Fu([2\()p Fj(n)h Fu(+)f(1\))g(+)g Fj(q)342 b Fd(\241)296 b Fu(1]!)p 8434 17471 10275 54 v 8582 18688 a([)q Fj(q)342 b Fd(\241)296 b Fu(1]!)221 b([)q(2\()p Fj(n)296 b Fu(+)f(1\)]!)18842 15903 y Fh(\276)221 b(\267)20893 16878 y Fu(2\()p Fj(n)296 b Fu(+)f(1\))p 20893 17471 4691 54 v 22926 18688 a Fj(q)25717 15903 y Fh(\270)26787 17776 y Fu(=)28167 15903 y Fh(\275)29296 16878 y Fu(\(2)p Fj(n)h Fu(+)f Fj(q)343 b Fu(+)294 b(1\))q(!)p 29296 17471 7279 54 v 29806 18688 a Fj(q)48 b Fu(!)221 b(\(2)p Fj(n)296 b Fu(+)f(1\)!)36708 15903 y Fh(\276)38073 17776 y Fu(=)39453 15903 y Fh(\265)40431 16878 y Fu(2)p Fj(n)h Fu(+)f Fj(q)343 b Fu(+)294 b(1)43072 18688 y Fj(q)46337 15903 y Fh(\266)47315 17776 y Fj(:)800 21474 y Fu(By)434 b(using)g(the)f(recursiv)-36 b(e)433 b(de\257nition)g(\()p 0 1 0 0 TeXcolorcmyk(2)p (#equation.2) [[254 524 260 536] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))g(of)i(Catalan)f(n)-36 b(um)g(b)36 b(ers,)433 b(\()p 0 1 0 0 TeXcolorcmyk(12)p (#equation.12) [[381 524 392 536] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))h(b)36 b(ecomes)433 b(an)h(inden)-36 b(tit)g(y)-108 b(.)800 25911 y Fl(3)2152 b(Some)717 b(features)f(of)g(the)h(arra)-60 b(y)717 b Fk(g)69 b Fi(\()p Fk(n;)319 b(q)69 b Fi(\))800 28831 y Fu(The)434 b(sequence)16240 31126 y Fj(g)48 b Fu(\()p Fj(n;)221 b(q)48 b Fu(\))368 b(=)23043 30228 y(2)p Fj(n)296 b Fu(+)f Fj(q)48 b Fu(!)p 21788 30821 6525 54 v 21788 32038 a Fj(q)g Fu(!)p Fj(n)p Fu(!\()p Fj(n)295 b Fu(+)g(1\)!)28814 31126 y(=)369 b Fj(C)31126 31325 y Fm(n)31752 29253 y Fh(\265)32730 30228 y Fu(2)p Fj(n)296 b Fu(+)e Fj(q)34244 32038 y(q)36383 29253 y Fh(\266)800 34141 y Fu(can)333 b(b)36 b(e)333 b(seen)g(as)g(a)h (generalization)g(of)g(the)f(Catalan)h(sequence,)353 b(as)334 b(it)f(reduces)f(to)h(the)g(Catalan)h(sequence)800 35746 y(for)524 b Fj(q)570 b Fu(=)521 b(0.)849 b(There)523 b(are)h(also)h(some)e(other)h(in)-36 b(teresting)523 b(features.)848 b(In)523 b(T)-108 b(able)524 b(3)g(I)g(rep)36 b(ort)523 b(the)g(kno)-36 b(wn)800 37351 y(names)384 b(of)g(the)g(in)-36 b(teger)383 b(sequences,)394 b(referenced)384 b(in)f Fc(The)419 b(On-line)f(Encyclop)-66 b(e)g(dia)418 b(of)h(Inte)-66 b(ger)417 b(Se)-66 b(quenc)g(es)800 38956 y Fu([)p 0 1 0 0 TeXcolorcmyk(2)p (#cite.2) [[82 366 88 378] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(],)434 b(that)f(can)g(b)36 b(e)434 b(extracted)f(from)h(the)f (matrix)i Fj(g)48 b Fu(\()p Fj(n;)221 b(q)48 b Fu(\).)p 4171 40772 47472 45 v 4149 42377 45 1606 v 10354 42377 V 15363 42377 V 16050 41895 a(A000108)p 21569 42377 V 1330 w(A001700)p 27774 42377 V 1331 w(A002457)p 33980 42377 V 1330 w(A002802)p 40185 42377 V 1330 w(A002803)p 46390 42377 V 2559 w(none)p 51621 42377 V 4171 42421 47472 45 v 4149 44026 45 1606 v 10354 44026 V 11041 43545 a Fj(m)368 b Ff(\302)h Fj(q)p 15363 44026 V 5506 w Fe(0)p 21569 44026 V 5458 w(1)p 27774 44026 V 5458 w(2)p 33980 44026 V 5459 w(3)p 40185 44026 V 5458 w(4)p 46390 44026 V 4484 w(5)p 51621 44026 V 4171 44070 47472 45 v 4149 45676 45 1606 v 7057 45194 a Fu(-)p 10354 45676 V 6483 w Fe(0)p 15363 45676 V 5556 w Fu(1)p 21569 45676 V 5555 w(1)p 27774 45676 V 5555 w(1)p 33980 45676 V 5556 w(1)p 40185 45676 V 5555 w(1)p 46390 45676 V 4580 w(1)p 51621 45676 V 4149 47281 V 4835 46799 a(A000217)p 10354 47281 V 4264 w Fe(1)p 15363 47281 V 5556 w Fu(1)p 21569 47281 V 5555 w(3)p 27774 47281 V 5555 w(6)p 33980 47281 V 4905 w(10)p 40185 47281 V 4906 w(15)p 46390 47281 V 3930 w(21)p 51621 47281 V 4149 48886 V 4835 48404 a(A034827)p 10354 48886 V 4264 w Fe(2)p 15363 48886 V 5556 w Fu(2)p 21569 48886 V 4905 w(10)p 27774 48886 V 4905 w(30)p 33980 48886 V 4905 w(70)p 40185 48886 V 4255 w(140)p 46390 48886 V 3281 w(252)p 51621 48886 V 4149 50491 V 5937 50009 a(none)p 10354 50491 V 5363 w Fe(3)p 15363 50491 V 5556 w Fu(5)p 21569 50491 V 4905 w(35)p 27774 50491 V 4255 w(140)p 33980 50491 V 4255 w(420)p 40185 50491 V 3605 w(1050)p 46390 50491 V 2630 w(2310)p 51621 50491 V 4149 52096 V 5937 51614 a(none)p 10354 52096 V 5363 w Fe(4)p 15363 52096 V 4905 w Fu(14)p 21569 52096 V 4255 w(126)p 27774 52096 V 4256 w(630)p 33980 52096 V 3605 w(2310)p 40185 52096 V 3605 w(6930)p 46390 52096 V 1980 w(18018)p 51621 52096 V 4149 53701 V 7057 53219 a(-)p 10354 53701 V 6483 w Fe(5)p 15363 53701 V 4905 w Fu(42)p 21569 53701 V 4255 w(462)p 27774 53701 V 3605 w(2772)p 33980 53701 V 2955 w(12012)p 40185 53701 V 2956 w(42042)p 46390 53701 V 1330 w(126126)p 51621 53701 V 4149 55306 V 7057 54825 a(-)p 10354 55306 V 6483 w Fe(6)p 15363 55306 V 4255 w Fu(132)p 21569 55306 V 3605 w(1716)p 27774 55306 V 2955 w(12012)p 33980 55306 V 2955 w(60060)p 40185 55306 V 2305 w(240240)p 46390 55306 V 1331 w(816816)p 51621 55306 V 4171 55350 47472 45 v 10646 58727 a(T)-108 b(able)433 b(3:)i Fj(g)48 b Fu(\()p Fj(m;)221 b(q)48 b Fu(\))432 b(n)-36 b(um)g(b)36 b(ers)432 b(and)h(names)g(of)i(kno)-36 b(wn)434 b(sequences)2751 61937 y(The)379 b(\257rst)e(\257v)-36 b(e)379 b(columns)f(corresp)36 b(ond)378 b(to)g(the)g(kno)-36 b(wn)379 b(sequences:)551 b(A000108)380 b(\(Catalan\),)391 b(A001700,)800 63542 y(A002457,)695 b(A002802,)f(A002803.)1201 b(The)641 b(\257rst)e(t)-36 b(w)g(o)641 b(ro)-36 b(ws)641 b(corresp)36 b(ond)640 b(to)h(the)f(sequences)g(A000217)800 65147 y(\()p Fj(g)48 b Fu(\(1)p Fj(;)221 b(q)48 b Fu(\),)374 b(triangular)359 b(n)-36 b(um)g(b)36 b(ers\))357 b(and)i(A034827.)556 b(F)-108 b(or)358 b(the)h(other)g(ro)-36 b(ws)360 b(and)e(columns)h(no) g(reference)g(w)-36 b(as)800 66752 y(found)289 b(b)-36 b(y)289 b(the)f(author.)530 b(Also)290 b(the)e(sequence)h(on)g(the)g (main)g(diagonal)h Fj(g)48 b Fu(\()p Fj(k)d(;)221 b(k)45 b Fu(\))289 b(\(1,3,30,420,6930,126126,)p Fj(:)221 b(:)g(:)9 b Fu(\))800 68357 y(is)480 b(kno)-36 b(wn)481 b(as)f(A007004)i(and)d (the)h(sequence)g(on)g(the)f(diagonal)i Fj(g)48 b Fu(\()p Fj(k)d(;)221 b(k)372 b Fu(+)326 b(1\))480 b(\(1,6,70,1050,18018,)498 b Fj(:)221 b(:)g(:)q Fu(\))800 69962 y(is)434 b(kno)-36 b(wn)434 b(as)g(A024489.)p Black Black eop %%Page: 5 5 5 4 bop Black 0 TeXcolorgray Black 52150 -2672 a Fu(5)p Black 800 1424 a Fl(4)2152 b(Generating)716 b(function)800 4345 y Fu(Consider)434 b(the)f(algebraic)h(equation)g(in)g Fj(J)123 b Fu(:)19137 7278 y Fd(\241)p Fj(xJ)21751 6730 y Fn(2)22573 7278 y Fu(+)294 b(\(1)i Fd(\241)f Fj(y)48 b(x)p Fu(\))p Fj(J)418 b Fd(\241)295 b Fu(1)370 b(=)e(0)p Fj(;)16026 b Fu(\(13\))800 10212 y(and)433 b(its)h(solutions)15731 13284 y Fj(J)123 b Fu(\()p Fj(x;)221 b(y)48 b Fu(\))370 b(=)21471 12073 y(\(1)295 b Fd(\241)h Fj(y)48 b(x)p Fu(\))295 b Fd(\247)27801 10428 y Fh(q)p 29129 10428 8246 54 v 1645 x Fu(\(1)h Fd(\241)f Fj(y)48 b(x)p Fu(\))33836 11427 y Fn(2)34657 12073 y Fd(\241)295 b Fu(4)p Fj(x)p 21471 12979 15904 54 v 28728 14196 a Fu(2)p Fj(x)37507 13284 y(:)12620 b Fu(\(14\))800 16022 y(Let)481 b(no)-36 b(w)481 b(supp)36 b(ose)480 b(that)g Fj(J)123 b Fu(\()p Fj(x;)221 b(y)48 b Fu(\))482 b(admits)e(a)i(T)-108 b(a)-36 b(ylor)482 b(expansion)f(in)g Fj(x)g Fu(\(whic)-36 b(h)480 b(excludes)h(the)g (solution)800 18258 y Fj(J)123 b Fu(\()p Fj(x;)221 b(y)48 b Fu(\))369 b(=)6540 17532 y Fn(\(1)p Fg(\241)p Fm(y)32 b(x)p Fn(\)+)10233 16571 y Fd(p)p 11339 16571 5154 54 v 11339 17532 a Fn(\(1)p Fg(\241)p Fm(y)g(x)p Fn(\))14300 17142 y Fb(2)14761 17532 y Fg(\241)p Fn(4)p Fm(x)p 6540 17953 9954 54 v 11016 18716 a Fn(2)p Fm(x)17059 18258 y Fu(unlimited)433 b(in)h Fj(x)369 b Fu(=)f(0\))20094 22330 y Fj(J)123 b Fu(\()p Fj(x;)221 b(y)48 b Fu(\))369 b(=)26247 20669 y Fg(1)25758 21068 y Fh(X)25700 23857 y Fm(m)p Fn(=0)27956 22330 y Fj(j)28494 22529 y Fm(m)29604 22330 y Fu(\()o Fj(y)48 b Fu(\))221 b Fj(x)32257 21781 y Fm(m)33145 22330 y Fj(:)16982 b Fu(\(15\))800 26426 y(Substituting)432 b(\()p 0 1 0 0 TeXcolorcmyk(15)p (#equation.15) [[151 479 163 491] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))i(in)-36 b(to)433 b(equation)h(\()p 0 1 0 0 TeXcolorcmyk(13)p (#equation.13) [[247 479 259 491] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))g(w)-36 b(e)434 b(get)2688 29320 y(0)370 b(=)e Fd(\241)p Fj(x)7081 28324 y Fh(P)8484 28675 y Fg(1)8484 29708 y Fm(m)p Fn(=0)10795 29320 y Fj(j)11333 29519 y Fm(m)12443 29320 y Fu(\()o Fj(y)48 b Fu(\))221 b Fj(x)15096 28838 y Fm(m)16206 28324 y Fh(P)17608 28675 y Fg(1)17608 29708 y Fm(m)p Fn(=0)19919 29320 y Fj(j)20457 29519 y Fm(n)21305 29320 y Fu(\()p Fj(y)48 b Fu(\))221 b Fj(x)23959 28838 y Fm(n)24880 29320 y Fu(+)26187 28324 y Fh(P)27589 28675 y Fg(1)27589 29708 y Fm(m)p Fn(=0)29900 29320 y Fj(j)30438 29519 y Fm(m)31548 29320 y Fu(\()o Fj(y)48 b Fu(\))221 b Fj(x)34201 28838 y Fm(m)35384 29320 y Fd(\241)296 b Fj(y)48 b(x)38355 28324 y Fh(P)39757 28675 y Fg(1)39757 29708 y Fm(m)p Fn(=0)42069 29320 y Fj(j)42607 29519 y Fm(m)43716 29320 y Fu(\()o Fj(y)g Fu(\))222 b Fj(x)46370 28838 y Fm(m)47552 29320 y Fd(\241)296 b Fu(1)369 b(=)3659 30925 y(=)g Fd(\241)p Fj(x)7033 29929 y Fh(P)8436 30280 y Fg(1)8436 31313 y Fm(s)p Fn(=0)10350 30925 y Fj(j)10888 31124 y Fm(s)11600 30925 y Fu(\()o Fj(y)48 b Fu(\))13514 29929 y Fh(P)14917 30280 y Fg(1)14917 31313 y Fm(m)p Fn(=)p Fm(s)17193 30925 y Fj(j)17731 31124 y Fm(s)p Fg(\241)p Fm(m)20007 30925 y Fu(\()o Fj(y)g Fu(\))221 b Fj(x)22660 30443 y Fm(s)23446 30925 y Fu(+)24753 29929 y Fh(P)26155 30280 y Fg(1)26155 31313 y Fm(m)p Fn(=0)28467 30925 y Fj(j)29005 31124 y Fm(m)30114 30925 y Fu(\()o Fj(y)48 b Fu(\))222 b Fj(x)32768 30443 y Fm(m)33950 30925 y Fd(\241)296 b Fj(y)36182 29929 y Fh(P)37584 30280 y Fg(1)37584 31313 y Fm(m)p Fn(=0)39896 30925 y Fj(j)40434 31124 y Fm(m)41543 30925 y Fu(\()o Fj(y)48 b Fu(\))221 b Fj(x)44196 30443 y Fm(m)p Fn(+1)46582 30925 y Fd(\241)295 b Fu(1)369 b(=)4601 32531 y(=)g Fd(\241)7236 31534 y Fh(P)8639 31885 y Fg(1)8639 32918 y Fm(s)p Fn(=0)10553 31534 y Fh(P)11955 31885 y Fm(s)11955 32918 y(m)p Fn(=0)14266 32531 y Fj(j)14804 32730 y Fm(s)15516 32531 y Fu(\()p Fj(y)48 b Fu(\))221 b Fj(j)17969 32730 y Fm(s)p Fg(\241)p Fm(m)20245 32531 y Fu(\()p Fj(y)48 b Fu(\))221 b Fj(x)22899 32048 y Fm(s)p Fn(+1)24887 32531 y Fu(+)26194 31534 y Fh(P)27596 31885 y Fg(1)27596 32918 y Fm(s)p Fn(=0)29510 32531 y Fj(j)30048 32730 y Fm(s)30760 32531 y Fu(\()p Fj(y)48 b Fu(\))221 b Fj(x)33414 32048 y Fm(s)34200 32531 y Fd(\241)295 b Fj(y)36431 31534 y Fh(P)37833 31885 y Fg(1)37833 32918 y Fm(s)p Fn(=0)39748 32531 y Fj(j)40286 32730 y Fm(s)40998 32531 y Fu(\()o Fj(y)48 b Fu(\))221 b Fj(x)43651 32048 y Fm(s)p Fn(+1)45639 32531 y Fd(\241)296 b Fu(1)369 b(=)6847 34136 y(=)f Fj(j)8765 34335 y Fn(0)9291 34136 y Fu(\()p Fj(y)48 b Fu(\))295 b Fd(\241)g Fu(1)h(+)14861 33139 y Fh(P)16263 33490 y Fg(1)16263 34523 y Fm(s)p Fn(=0)18177 34136 y Fu([)p Fd(\241)19792 33139 y Fh(P)21195 33490 y Fm(s)21195 34523 y(m)p Fn(=0)23506 34136 y Fj(j)24044 34335 y Fm(s)24756 34136 y Fu(\()p Fj(y)48 b Fu(\))221 b Fj(j)27209 34335 y Fm(s)p Fg(\241)p Fm(m)29485 34136 y Fu(\()p Fj(y)48 b Fu(\))295 b(+)g Fj(j)33319 34335 y Fm(s)p Fn(+1)35233 34136 y Fu(\()p Fj(y)48 b Fu(\))295 b Fd(\241)g Fj(y)48 b(j)39770 34335 y Fm(s)p Fg(\241)p Fn(1)41684 34136 y Fu(\()p Fj(y)g Fu(\))o(])222 b Fj(x)44699 33654 y Fm(s)p Fn(+1)46392 34136 y Fj(:)800 36979 y Fu(Then)24221 38213 y Fj(j)24759 38412 y Fn(0)25285 38213 y Fu(\()p Fj(y)48 b Fu(\))369 b(=)f(1)15038 39818 y Fj(j)15576 40017 y Fm(s)p Fn(+1)17490 39818 y Fu(\()p Fj(y)48 b Fu(\))368 b(=)h Fj(y)481 b(j)22586 40017 y Fm(s)23298 39818 y Fu(\()p Fj(y)48 b Fu(\))295 b(+)26594 38822 y Fh(P)27996 39172 y Fm(s)27996 40205 y(m)p Fn(=0)30307 39818 y Fj(j)30845 40017 y Fm(s)31557 39818 y Fu(\()p Fj(y)48 b Fu(\))221 b Fj(j)34010 40017 y Fm(s)p Fg(\241)p Fm(m)36286 39818 y Fu(\()p Fj(y)48 b Fu(\))221 b Fj(;)50488 39027 y Fu(\(16\))800 42126 y(whic)-36 b(h)433 b(is)h(the)f(recursiv) -36 b(e)434 b(de\257nition)f(giv)-36 b(en)434 b(b)-36 b(y)433 b(\()p 0 1 0 0 TeXcolorcmyk(1)p (#equation.1) [[297 338 303 350] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\).)579 b(This)434 b(means)f(that)19378 45792 y Fj(j)19916 45991 y Fm(m)21025 45792 y Fu(\()p Fj(y)48 b Fu(\))369 b(=)436 b(lim)24468 46589 y Fm(x)p Fg(!)p Fn(0)26967 44893 y Fu(1)p 26542 45487 1499 54 v 26542 46703 a Fj(m)p Fu(!)28307 44893 y Fj(d)28983 44411 y Fm(m)29870 44893 y Fj(J)123 b Fu(\()p Fj(x;)221 b(y)48 b Fu(\))p 28307 45487 5421 54 v 29866 46703 a Fj(dx)31281 46319 y Fm(m)33861 45792 y Fj(;)800 49157 y Fu(and)433 b(for)h Fj(y)417 b Fu(=)369 b(0)434 b(the)f(function)g Fj(J)123 b Fu(\()p Fj(x;)221 b Fu(0\))434 b(is)g(the)f(generating)h (function)f(of)h(the)f(Catalan)i(sequence)23472 52052 y Fj(j)24010 52251 y Fm(m)24898 52052 y Fu(\(0\))369 b(=)g Fj(C)29241 52251 y Fm(m)19112 53871 y Fj(J)123 b Fu(\()p Fj(x;)221 b Fu(0\))369 b(=)24820 53300 y Fn(1)p Fg(\241)26022 52613 y(p)p 26806 52613 2203 40 v 687 x Fn(1)p Fg(\241)p Fn(4)p Fm(x)p 24820 53565 4189 54 v 26414 54329 a Fn(2)p Fm(x)29510 53871 y Fu(=)g Fj(C)31822 54070 y Fm(a)32377 53871 y Fu(\()p Fj(x)p Fu(\))p Fj(:)800 56787 y Fu(No)-36 b(w,)435 b(using)e(Equations)h(\()p 0 1 0 0 TeXcolorcmyk(4)p (#equation.4) [[199 206 205 218] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))f(and)g(\()p 0 1 0 0 TeXcolorcmyk(15)p (#equation.15) [[241 206 253 218] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\),)i(w)-36 b(e)433 b(get)8186 59682 y Fj(J)123 b Fu(\()p Fj(x;)221 b(y)48 b Fu(\))369 b(=)13793 58686 y Fh(P)15195 59036 y Fg(1)15195 60070 y Fm(m)p Fn(=0)17506 58686 y Fh(P)18908 59036 y Fm(m)18908 60070 y(q)32 b Fn(=0)20839 59682 y Fj(e)p Fu(\()p Fj(m;)221 b(q)48 b Fu(\))433 b Fj(y)25914 59200 y Fm(q)26421 59682 y Fj(x)27160 59200 y Fm(m)28417 59682 y Fu(=)29797 58686 y Fh(P)31199 59036 y Fg(1)31199 60070 y Fm(q)32 b Fn(=0)33130 58686 y Fh(P)34532 59036 y Fg(1)34532 60070 y Fm(m)p Fn(=)p Fm(q)36825 59682 y Fj(e)p Fu(\()p Fj(m;)221 b(q)48 b Fu(\))433 b Fj(y)41900 59200 y Fm(q)42407 59682 y Fj(x)43146 59200 y Fm(m)44403 59682 y Fu(=)7964 61365 y(=)9344 60369 y Fh(P)10746 60720 y Fg(1)10746 61753 y Fm(q)32 b Fn(=0)12677 61365 y Fj(y)13359 60883 y Fm(q)14088 60369 y Fh(P)15490 60720 y Fg(1)15490 61753 y Fm(m)p Fn(=0)17801 61365 y Fj(e)p Fu(\()p Fj(m)295 b Fu(+)g Fj(q)48 b(;)221 b(q)48 b Fu(\))432 b Fj(x)25159 60883 y Fm(m)p Fn(+)p Fm(q)27600 61365 y Fu(=)28980 60369 y Fh(P)30382 60720 y Fg(1)30382 61753 y Fm(q)32 b Fn(=0)32313 61365 y Fu(\()p Fj(y)48 b(x)p Fu(\))34746 60720 y Fm(q)35475 60369 y Fh(P)36877 60720 y Fg(1)36877 61753 y Fm(m)p Fn(=0)39188 61365 y Fj(g)g Fu(\()p Fj(m;)221 b(q)48 b Fu(\))432 b Fj(x)44387 60883 y Fm(m)45275 61365 y Fj(:)50488 60574 y Fu(\(17\))800 64287 y(Then,)i(the)f(function)13168 68946 y Fj(L)p Fu(\()p Fj(x;)221 b(z)59 b Fu(\))370 b(=)18933 67734 y(\(1)295 b Fd(\241)g Fj(z)59 b Fu(\))296 b Fd(\241)24505 66090 y Fh(q)p 25833 66090 7488 54 v 1644 x Fu(\(1)g Fd(\241)f Fj(z)59 b Fu(\))29782 67089 y Fn(2)30603 67734 y Fd(\241)296 b Fu(4)p Fj(x)p 18933 68641 14389 54 v 25432 69857 a Fu(2)p Fj(x)33823 68946 y Fu(=)369 b Fj(J)123 b Fu(\()p Fj(x;)221 b(z)59 b(=x)p Fu(\))p Black Black eop %%Page: 6 6 6 5 bop Black 0 TeXcolorgray Black 52150 -2672 a Fu(6)p Black 800 1424 a(can)434 b(b)36 b(e)433 b(expanded)g(to)h(get)f(\(see)h (equation)g(\()p 0 1 0 0 TeXcolorcmyk(17)p (#equation.17) [[279 704 290 716] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\)\))17977 5299 y Fj(L)p Fu(\()p Fj(x;)221 b(z)59 b Fu(\))370 b(=)24097 3639 y Fg(1)23608 4037 y Fh(X)23740 6827 y Fm(q)32 b Fn(=0)26295 3639 y Fg(1)25806 4037 y Fh(X)25748 6827 y Fm(m)p Fn(=0)28004 5299 y Fj(g)48 b Fu(\()p Fj(m;)221 b(q)48 b Fu(\))433 b Fj(x)33204 4751 y Fm(m)34091 5299 y Fj(z)34754 4751 y Fm(q)35262 5299 y Fj(;)14865 b Fu(\(18\))800 9496 y(and)433 b(this)g(can)h(b)36 b(e)433 b(seen)h(to)f(b)36 b(e)434 b(the)f(generating)h(function)f(of)h Fj(g)48 b Fu(\()p Fj(m;)221 b(q)48 b Fu(\):)17312 13112 y Fj(g)g Fu(\()p Fj(m;)221 b(q)48 b Fu(\))368 b(=)804 b(lim)23088 13909 y Fm(x;z)37 b Fg(!)p Fn(0)27249 12214 y Fu(1)p 26332 12807 2485 54 v 26332 14023 a Fj(m)p Fu(!)p Fj(q)48 b Fu(!)29082 12214 y Fj(@)29841 11732 y Fm(m)p Fn(+)p Fm(q)31912 12214 y Fj(L)p Fu(\()p Fj(x;)221 b(z)59 b Fu(\))p 29082 12807 6712 54 v 30281 14023 a Fj(@)72 b(x)31779 13640 y Fm(m)32666 14023 y Fj(@)g(z)34088 13640 y Fm(q)35927 13112 y Fj(:)2751 16572 y Fu(It)434 b(is)g(in)-36 b(teresting)433 b(to)g(observ)-36 b(e)434 b(that)8068 20374 y Fj(L)p Fu(\()p Fj(x;)221 b(z)59 b Fu(\))1108 b(=)15665 18713 y Fg(1)15176 19112 y Fh(X)15308 21901 y Fm(q)32 b Fn(=0)17863 18713 y Fg(1)17374 19112 y Fh(X)17316 21901 y Fm(m)p Fn(=0)19572 20374 y Fj(g)48 b Fu(\()p Fj(m;)221 b(q)48 b Fu(\))432 b Fj(x)24771 19825 y Fm(m)25659 20374 y Fj(z)26322 19825 y Fm(q)27198 20374 y Fu(=)29126 18713 y Fg(1)28637 19112 y Fh(X)28579 21901 y Fm(m)p Fn(=0)30835 20374 y Fj(C)31766 20573 y Fm(m)33364 18713 y Fg(1)32875 19112 y Fh(X)33007 21901 y Fm(q)32 b Fn(=0)35015 18501 y Fh(\265)35993 19475 y Fu(2)p Fj(m)295 b Fu(+)g Fj(q)37688 21285 y(q)40007 18501 y Fh(\266)40985 20374 y Fj(z)41648 19825 y Fm(q)42155 20374 y Fj(x)42894 19825 y Fm(m)44151 20374 y Fu(=)13057 24824 y(=)15723 23163 y Fg(1)15234 23562 y Fh(X)15176 26351 y Fm(m)p Fn(=0)17432 24824 y Fj(C)18363 25023 y Fm(m)19250 24824 y Fj(T)20012 25023 y Fm(m)20900 24824 y Fu(\()p Fj(z)59 b Fu(\))p Fj(x)23314 24275 y Fm(m)800 28775 y Fu(with)19442 31060 y Fj(T)20204 31259 y Fm(m)21092 31060 y Fu(\()p Fj(z)g Fu(\))369 b(=)25005 29400 y Fg(1)24516 29798 y Fh(X)24649 32588 y Fm(q)32 b Fn(=0)26656 29187 y Fh(\265)27634 30161 y Fu(2)p Fj(m)295 b Fu(+)g Fj(q)29329 31971 y(q)31649 29187 y Fh(\266)32626 31060 y Fj(z)33289 30512 y Fm(q)33797 31060 y Fj(:)800 34615 y Fu(It)434 b(can)f(b)36 b(e)434 b(pro)-36 b(v)g(en)433 b(that)20875 36739 y Fj(T)21637 36938 y Fm(m)22525 36739 y Fu(\()p Fj(z)59 b Fu(\))369 b(=)29012 35840 y(1)p 26083 36433 6510 54 v 26083 37650 a(\(1)295 b Fd(\241)g Fj(z)59 b Fu(\))30031 37266 y Fn(2)p Fm(m)p Fn(+1)800 39354 y Fu(as)17045 40461 y(1)p 16877 41054 986 54 v 16877 42271 a Fj(q)48 b Fu(!)18129 40461 y Fj(d)18805 39979 y Fm(q)19312 40461 y Fj(T)20074 40660 y Fm(m)20962 40461 y Fu(\()p Fj(z)59 b Fu(\))p 18129 41054 4509 54 v 19460 42271 a Fj(dz)20799 41887 y Fm(q)23139 41359 y Fu(=)24652 40461 y(\(2)p Fj(m)295 b Fu(+)g(1\))221 b Fd(\242)g(\242)g(\242)i Fu(\(2)p Fj(m)295 b Fu(+)f Fj(q)48 b Fu(\))p 24652 41054 12071 54 v 26131 42271 a Fj(q)g Fu(!)433 b(\(1)296 b Fd(\241)f Fj(z)59 b Fu(\))31499 41887 y Fn(2)p Fm(m)p Fn(+1+)p Fm(q)800 44325 y Fu(and)15143 46362 y(lim)15103 47159 y Fm(z)37 b Fg(!)p Fn(0)17290 45463 y Fu(1)p 17122 46056 986 54 v 17122 47273 a Fj(q)48 b Fu(!)18373 45463 y Fj(d)19049 44981 y Fm(q)19556 45463 y Fj(T)20318 45662 y Fm(m)21206 45463 y Fu(\()p Fj(z)59 b Fu(\))p 18373 46056 4509 54 v 19704 47273 a Fj(dz)21043 46889 y Fm(q)23383 46362 y Fu(=)24896 45463 y(\(2)p Fj(m)296 b Fu(+)e Fj(q)48 b Fu(\)!)p 24896 46056 5388 54 v 25806 47273 a Fj(q)g Fu(!)433 b(2)p Fj(m)p Fu(!)30786 46362 y(=)32166 44489 y Fh(\265)33144 45463 y Fu(2)p Fj(m)295 b Fu(+)g Fj(q)34839 47273 y(q)37158 44489 y Fh(\266)38136 46362 y Fj(:)2751 49420 y Fu(The)434 b(generating)f(function)h Fj(L)p Fu(\()p Fj(x;)221 b(z)59 b Fu(\))434 b(can)g(then)e(b)36 b(e)434 b(written)f(also)h(in)g(the)f(form)9516 53295 y Fj(L)p Fu(\()p Fj(x;)221 b(z)59 b Fu(\))370 b(=)16929 52396 y(1)p 15280 52989 3949 54 v 15280 54206 a(\(1)295 b Fd(\241)h Fj(z)59 b Fu(\))20072 51634 y Fg(1)19583 52033 y Fh(X)19656 54822 y Fm(n)p Fn(=0)21723 53295 y Fj(C)22654 53494 y Fm(n)23502 51422 y Fh(\267)26203 52396 y Fj(x)p 24336 52989 4475 54 v 24336 54206 a Fu(\(1)295 b Fd(\241)g Fj(z)59 b Fu(\))28284 53822 y Fn(2)28943 51422 y Fh(\270)29644 51719 y Fm(n)30639 53295 y Fu(=)33802 52396 y(1)p 32153 52989 3949 54 v 32153 54206 a(\(1)295 b Fd(\241)g Fj(z)59 b Fu(\))36235 53295 y Fj(C)37166 53494 y Fm(a)37942 51422 y Fh(\267)40643 52396 y Fj(x)p 38776 52989 4475 54 v 38776 54206 a Fu(\(1)295 b Fd(\241)g Fj(z)59 b Fu(\))42724 53822 y Fn(2)43383 51422 y Fh(\270)800 57320 y Fu(that)332 b(b)36 b(etter)332 b(sho)-36 b(ws)332 b(the)g(strong)h(link)g(existing)g(b)36 b(et)-36 b(w)g(een)332 b(the)g(sequence)g(arra)-36 b(y)333 b Fj(g)48 b Fu(\()p Fj(n;)221 b(q)48 b Fu(\))332 b(and)g(the)g(Catalan)800 58925 y(n)-36 b(um)g(b)36 b(ers.)800 63351 y Fl(5)2152 b(App)60 b(endix)800 66272 y Fu(The)434 b(follo)-36 b(wing)435 b(binomial)g(iden)-36 b(tit)g(y:)14322 68413 y Fm(q)13589 68865 y Fh(X)13710 71655 y Fm(p)p Fn(=0)15729 68254 y Fh(\265)16707 69229 y Fj(m)294 b Fu(+)h Fj(p)17834 71038 y(m)20099 68254 y Fh(\266\265)22055 69229 y Fj(n)g Fu(+)g Fj(q)343 b Fd(\241)295 b Fj(p)24306 71038 y(n)27334 68254 y Fh(\266)28681 70127 y Fu(=)30061 68254 y Fh(\265)31039 69229 y Fj(m)g Fu(+)g Fj(n)g Fu(+)g Fj(q)343 b Fu(+)294 b(1)34724 71038 y Fj(q)39034 68254 y Fh(\266)50488 70127 y Fu(\(19\))p Black Black eop %%Page: 7 7 7 6 bop Black 0 TeXcolorgray Black 52150 -2672 a Fu(7)p Black 800 1424 a(holds)433 b(for)i(an)-36 b(y)433 b(non-negativ)-36 b(e)434 b(in)-36 b(tegers)433 b Fj(m;)221 b(n;)g(q)48 b Fu(.)800 4358 y Fe(Pro)42 b(of.)578 b Fu(W)-108 b(e)434 b(de\257ne)15327 6697 y Fj(M)139 b Fu(\()p Fj(m;)221 b(n;)g(q)48 b Fu(\))369 b(=)23922 4982 y Fm(q)23189 5435 y Fh(X)23310 8225 y Fm(p)p Fn(=0)25329 4824 y Fh(\265)26307 5798 y Fj(m)295 b Fu(+)g Fj(p)27434 7608 y(m)29699 4824 y Fh(\266\265)31655 5798 y Fj(n)h Fu(+)e Fj(q)343 b Fd(\241)295 b Fj(p)33906 7608 y(n)36934 4824 y Fh(\266)37912 6697 y Fj(:)800 10356 y Fu(Then)433 b(the)g(prop)36 b(osition)434 b(\()p 0 1 0 0 TeXcolorcmyk(19)p (#equation.19) [[196 624 208 636] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))g(is)g(equiv)-72 b(alen)-36 b(t)434 b(to)17894 14053 y Fj(M)139 b Fu(\()p Fj(m;)221 b(n;)g(q)48 b Fu(\))369 b(=)25756 12180 y Fh(\265)26734 13155 y Fj(m)295 b Fu(+)f Fj(n)i Fu(+)f Fj(q)342 b Fu(+)295 b(1)30419 14965 y Fj(q)34728 12180 y Fh(\266)800 17677 y Fu(The)434 b(pro)36 b(of)434 b(is)g(based)f(on)g(the)g(use)h(of)g(the)f(binomial)h(iden)-36 b(tit)g(y)21699 19888 y Fm(n)21025 20287 y Fh(X)21136 23116 y Fm(k)24 b Fn(=)p Fm(r)23165 19676 y Fh(\265)24143 20650 y Fj(k)24192 22460 y(r)24864 19676 y Fh(\266)26210 21549 y Fu(=)27591 19676 y Fh(\265)28569 20650 y Fj(n)295 b Fu(+)g(1)28646 22460 y Fj(r)331 b Fu(+)295 b(1)31598 19676 y Fh(\266)50488 21549 y Fu(\(20\))800 25610 y(that)433 b(can)h(also)g(b)36 b(e)434 b(written)f(as)18982 27563 y Fm(m)18439 27962 y Fh(X)18541 30791 y Fm(k)24 b Fn(=0)20579 27351 y Fh(\265)21557 28325 y Fj(r)331 b Fu(+)295 b Fj(k)22719 30135 y(r)24502 27351 y Fh(\266)25849 29224 y Fu(=)27230 27351 y Fh(\265)28208 28325 y Fj(m)f Fu(+)h Fj(r)332 b Fu(+)295 b(1)29577 30135 y Fj(r)332 b Fu(+)295 b(1)33822 27351 y Fh(\266)34800 29224 y Fj(:)800 33359 y Fu(The)434 b(iden)-36 b(tit)g(y)433 b(\()p 0 1 0 0 TeXcolorcmyk(19)p (#equation.19) [[151 417 163 429] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))h(holds)f(for)h Fj(n)370 b Fu(=)e(0)434 b(and)f(an)-36 b(y)434 b Fj(m;)221 b(q)48 b Fu(.)578 b(In)433 b(fact,)2639 37359 y Fj(M)139 b Fu(\()p Fj(m;)221 b Fu(0)p Fj(;)g(q)48 b Fu(\))369 b(=)11108 35644 y Fm(q)10375 36097 y Fh(X)10496 38886 y Fm(p)p Fn(=0)12515 35486 y Fh(\265)13493 36460 y Fj(m)294 b Fu(+)h Fj(p)14620 38270 y(m)16885 35486 y Fh(\266\265)18841 36460 y Fj(q)342 b Fd(\241)296 b Fj(p)19966 38270 y Fu(0)21741 35486 y Fh(\266)23088 37359 y Fu(=)25202 35644 y Fm(q)24469 36097 y Fh(X)24590 38886 y Fm(p)p Fn(=0)26609 35486 y Fh(\265)27587 36460 y Fj(m)f Fu(+)f Fj(p)28714 38270 y(m)30979 35486 y Fh(\266)32326 37359 y Fu(=)33707 35486 y Fh(\265)34684 36460 y Fj(m)h Fu(+)g Fj(q)343 b Fu(+)294 b(1)35798 38270 y Fj(m)g Fu(+)h(1)40301 35486 y Fh(\266)41648 37359 y Fu(=)43028 35486 y Fh(\265)44006 36460 y Fj(m)g Fu(+)g Fj(q)342 b Fu(+)295 b(1)46502 38270 y Fj(q)49622 35486 y Fh(\266)50600 37359 y Fj(;)800 41627 y Fu(where)434 b(the)f(binomial)h(iden)-36 b(tit)g(y)433 b(\()p 0 1 0 0 TeXcolorcmyk(20)p (#equation.20) [[229 342 241 354] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))h(w)-36 b(as)434 b(used.)2751 43232 y(F)-108 b(or)433 b(an)-36 b(y)434 b Fj(n)369 b Fd(6)p Fu(=)g(0,)434 b(using)g(\()p 0 1 0 0 TeXcolorcmyk(20)p (#equation.20) [[211 328 222 340] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))f(again,)i(and)e(with)h Fj(q)48 b(;)221 b(m)433 b Fu(natural)g(n)-36 b(um)g(b)36 b(ers,)2597 46929 y Fj(M)139 b Fu(\()p Fj(m;)221 b(n;)g(q)48 b Fu(\))370 b(=)10459 45933 y Fh(P)11862 46283 y Fm(q)11862 47316 y(p)p Fn(=0)13814 45056 y Fh(\265)14792 46030 y Fj(m)295 b Fu(+)g Fj(p)15919 47840 y(m)18185 45056 y Fh(\266)o(\265)20140 46030 y Fj(n)h Fu(+)f Fj(q)342 b Fd(\241)295 b Fj(p)22392 47840 y(n)25419 45056 y Fh(\266)26766 46929 y Fu(=)28147 45933 y Fh(P)29549 46283 y Fm(q)29549 47316 y(p)p Fn(=0)31502 45056 y Fh(\265)32479 46030 y Fj(m)g Fu(+)g Fj(p)33607 47840 y(m)35872 45056 y Fh(\266\265)37827 46030 y Fu(\()p Fj(n)h Fd(\241)f Fu(1\))h(+)e Fj(q)343 b Fd(\241)295 b Fj(p)g Fu(+)g(1)40079 47840 y(\()p Fj(n)g Fd(\241)h Fu(1\))f(+)g(1)48644 45056 y Fh(\266)49991 46929 y Fu(=)4997 50117 y(=)6377 49121 y Fh(P)7779 49471 y Fm(q)7779 50504 y(p)p Fn(=0)9732 48244 y Fh(\265)10710 49218 y Fj(m)g Fu(+)g Fj(p)11837 51028 y(m)14102 48244 y Fh(\266)15302 49121 y(P)16704 49471 y Fm(q)32 b Fg(\241)p Fm(p)16704 50504 y(k)24 b Fn(=0)18696 48244 y Fh(\265)19674 49218 y Fj(n)296 b Fd(\241)f Fu(1)h(+)e Fj(k)20836 51028 y(n)h Fd(\241)h Fu(1)25047 48244 y Fh(\266)26394 50117 y Fu(=)27775 49121 y Fh(P)29177 49471 y Fm(q)29177 50504 y(k)24 b Fn(=0)31170 49121 y Fh(P)32572 49471 y Fm(q)32 b Fg(\241)p Fm(k)32572 50504 y(p)p Fn(=0)34546 48244 y Fh(\265)35524 49218 y Fj(m)294 b Fu(+)h Fj(p)36651 51028 y(m)38916 48244 y Fh(\266\265)40872 49218 y Fj(n)g Fd(\241)h Fu(1)f(+)g Fj(k)42033 51028 y(n)h Fd(\241)f Fu(1)46245 48244 y Fh(\266)47592 50117 y Fu(=)9415 53305 y(=)10795 52309 y Fh(P)12198 52659 y Fm(q)12198 53692 y(k)24 b Fn(=0)14190 51432 y Fh(\265)15168 52406 y Fj(q)342 b Fd(\241)296 b Fj(k)340 b Fu(+)295 b Fj(m)g Fu(+)g(1)17453 54216 y Fj(m)g Fu(+)g(1)23129 51432 y Fh(\266\265)25085 52406 y Fj(n)g Fd(\241)g Fu(1)h(+)f Fj(k)26246 54216 y(n)h Fd(\241)f Fu(1)30458 51432 y Fh(\266)31805 53305 y Fu(=)368 b Fj(M)139 b Fu(\()p Fj(m)295 b Fu(+)g(1)p Fj(;)221 b(n)296 b Fd(\241)g Fu(1)p Fj(;)221 b(q)48 b Fu(\))p Fj(:)800 57002 y Fu(By)434 b(rep)36 b(eatedly)434 b(applying)g(the)f(rule)g Fj(M)139 b Fu(\()p Fj(m;)221 b(n;)g(q)48 b Fu(\))370 b(=)e Fj(M)139 b Fu(\()p Fj(m)295 b Fu(+)g(1)p Fj(;)221 b(n)297 b Fd(\241)e Fu(1)p Fj(;)221 b(q)48 b Fu(\),)434 b(it)g(is)g(easy)g(to)g(obtain)12656 60700 y Fj(M)139 b Fu(\()p Fj(m;)221 b(n;)g(q)48 b Fu(\))369 b(=)g Fj(M)139 b Fu(\()p Fj(m)295 b Fu(+)g Fj(n;)221 b Fu(0)p Fj(;)g(q)48 b Fu(\))370 b(=)30632 58827 y Fh(\265)31610 59801 y Fj(m)295 b Fu(+)g Fj(n)h Fu(+)e Fj(q)343 b Fu(+)295 b(1)35295 61611 y Fj(q)39605 58827 y Fh(\266)40583 60700 y Fj(:)p 1234 64051 651 651 v Black Black eop %%Page: 8 8 8 7 bop Black 0 TeXcolorgray Black 52150 -2672 a Fu(8)p Black 800 1424 a Fl(References)p Black 800 4345 a Fu([1])p Black 651 w(R.)434 b(P)-108 b(.)434 b(Stanley)-108 b(.)576 b Fc(Enumer)-66 b(ative)464 b(Combinatorics)p Fu(,)432 b(V)-108 b(ol.)434 b(2.)577 b(Cam)-36 b(bridge)434 b(Univ)-36 b(ersit)g(y)434 b(Press,)g(1999.)p Black 800 7057 a([2])p Black 651 w(N.)k(J.)g(A.)f(Sloane.)588 b Fc(On-Line)467 b(Encyclop)-66 b(e)g(dia)467 b(of)h(Inte)-66 b(ger)466 b(Se)-66 b(quenc)g(es)p Fu(.)586 b(published)436 b(electronically)j(at) p 0 1 0 0 TeXcolorcmyk 2823 8662 a Fa(http://www.research.att.com/)p Fd(\273)p Fa(njas/sequences)p [[97 639 365 651] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/~njas/sequences) pdfm Black Fu(.)p 800 11769 52000 45 v 800 14019 a(2000)391 b Fc(Mathematics)424 b(Subje)-66 b(ct)423 b(Classi\257c)-66 b(ation)p Fu(:)555 b(Primary)390 b(11B83;)406 b(Secondary)390 b(05A15,)400 b(11Y55,)g(11B65.)800 15625 y Fc(Keywor)-66 b(ds:)597 b(Gener)-66 b(ating)463 b(function,)g(Catalan)h(numb)-66 b(ers,)464 b(binomial)f(identity,)g(p)-66 b(olynomials)p 800 17255 V 800 19580 a Fu(\(Concerned)373 b(with)g(sequences)p 0 1 0 0 TeXcolorcmyk 373 w(A000108)p 16422 19793 4878 54 v [[220 541 264 553] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A000108) pdfm Black 0 1 0 0 TeXcolorcmyk 375 w(A000217)p 21672 19793 V [[267 541 311 553] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A000217) pdfm Black 0 1 0 0 TeXcolorcmyk 375 w(A034827)p 26922 19793 V [[314 541 358 553] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A034827) pdfm Black 0 1 0 0 TeXcolorcmyk 375 w(A001700)p 32172 19793 V [[362 541 405 553] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A001700) pdfm Black 0 1 0 0 TeXcolorcmyk 376 w(A002802)p 37423 19793 V [[409 541 453 553] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A002802) pdfm Black 0 1 0 0 TeXcolorcmyk 375 w(A002803)p 42673 19793 V [[456 541 500 553] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A002803) pdfm Black 0 1 0 0 TeXcolorcmyk 375 w(A007004)p 47923 19793 V [[503 541 547 553] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A007004) pdfm Black 0 1 0 0 TeXcolorcmyk 800 21185 a(A024489)p 800 21398 V [[79 526 123 538] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A024489) pdfm Black 0 1 0 0 TeXcolorcmyk 436 w(A002457)p 6111 21398 V [[127 526 171 538] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A002457) pdfm Black 2 w(.\))p 800 22890 52000 45 v 800 25941 a(Receiv)-36 b(ed)468 b(Octob)36 b(er)467 b(6,)476 b(2002;)486 b(revised)468 b(v)-36 b(ersion)468 b(receiv)-36 b(ed)467 b(April)h(17,)476 b(2003.)681 b(Published)467 b(in)g Fc(Journal)496 b(of)800 27546 y(Inte)-66 b(ger)463 b(Se)-66 b(quenc)g(es)431 b Fu(Ma)-36 b(y)434 b(2,)g(2003.)p 800 29177 V 800 31428 a(Return)f(to)p 0 1 0 0 TeXcolorcmyk 433 w(Journal)h(of)g(In)-36 b(teger)434 b(Sequences)f(home)g(page)p [[133 434 338 446] [1 1 1 [3 3]] [0 0 1]] (http://www.math.uwaterloo.ca/JIS/) pdfm Black(.)p Black Black eop %%Trailer end end