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(#equation.2) [[217 577 223 589] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))h(can)g(b)36 b(e)443 b(pro)-36 b(v)g(en)443 b(automatically)i(b)-36 b(y)443 b(computer,)i(using)e(the)g(metho)36 b(ds)800 17139 y(and)401 b(soft)-36 b(w)g(are)403 b(of)g(P)-36 b(etk)g(o)g(v)-69 b(\267)-582 b(sek,)410 b(Wilf)404 b(and)d(Zeilb)36 b(erger)402 b([)p 0 1 0 0 TeXcolorcmyk(5)p (#cite.5) [[324 563 330 575] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(].)568 b(W)-108 b(e)402 b(o\256er)f(an)h(indep)36 b(enden)-36 b(t,)407 b(non-automatic)800 18744 y(pro)36 b(of)409 b(that)f(in)-36 b(v)g(olv)g(es)409 b(a)f(certain)h(symmetry)f (idea)h(whic)-36 b(h)408 b(migh)-36 b(t)408 b(lead)g(to)g(the)g(disco) -36 b(v)g(ery)409 b(of)g(other)f(suc)-36 b(h)800 20349 y(iden)g(tities.)553 b(Tw)-36 b(o)359 b(simpler)f(auxiliary)h(iden)-36 b(tities)358 b(used)f(in)h(the)f(pro)36 b(of)359 b(are)f(also)h (automatically)h(v)-36 b(eri\257able)800 21954 y(and)433 b(shall)h(not)g(b)36 b(e)433 b(pro)-36 b(v)g(ed.)800 26340 y Fk(2)2152 b(Pro)60 b(of)716 b(of)h(the)g(main)g(result)800 29260 y Fd(Pro)42 b(of)480 b(of)616 b Fj(\()p 0 1 0 0 TeXcolorcmyk(1)p (#equation.1) [[137 454 143 466] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))p Fd(.)1301 b Fj(Our)417 b(pro)36 b(of)418 b(uses)g(a)g(v)-72 b(arian)-36 b(t)419 b(of)f(the)g(generating)g(function)g([)p 0 1 0 0 TeXcolorcmyk(6)p (#cite.6) [[430 454 436 466] [1 1 1 [3 3]] [0 0 1]] pdfm Black(])g(for)h(the)e(n)-36 b(um)g(b)36 b(ers)417 b Fi(m)51813 29459 y Fh(n)52439 29260 y Fj(,)800 30865 y(namely)9242 33409 y Fi(f)142 b Fj(\()p Fi(x)p Fj(\))369 b(=)13658 32510 y(1)296 b Fe(\241)f Fi(x)g Fj(+)18273 31418 y Fe(p)p 19380 31418 7181 54 v 1092 x Fj(1)h(+)e(2)p Fi(x)i Fe(\241)f Fj(3)p Fi(x)26034 32127 y Ff(2)p 13658 33103 12903 54 v 19784 34320 a Fj(2)27063 33409 y(=)368 b(1)296 b Fe(\241)f Fi(x)31456 32861 y Ff(2)32277 33409 y Fj(+)34073 31749 y Fc(1)33584 32147 y Fb(X)33657 34937 y Fh(n)p Ff(=3)35503 33409 y Fj(\()p Fe(\241)p Fj(1\))38198 32861 y Fh(n)p Ff(+1)40026 33409 y Fi(m)41164 33608 y Fh(n)p Fc(\241)p Ff(2)42993 33409 y Fi(x)43732 32861 y Fh(n)800 36748 y Fj(Then)475 b Fi(f)618 b Fj(satis\257es)476 b Fi(f)142 b Fj(\(0\))441 b(=)f(1)p Fi(;)221 b(f)142 b Fj(\(1\))442 b(=)f(0)476 b(and)f(is)h(decreasing)g(on)g([0)p Fi(;)221 b Fj(1].)707 b(Another)475 b(prop)36 b(ert)-36 b(y)475 b(of)i Fi(f)617 b Fj(that)800 38353 y(will)435 b(b)36 b(e)433 b(essen)-36 b(tial)434 b(in)g(the)f(pro)36 b(of)434 b(is)g(that)f(it)h(satis\257es)f(the)g(functional)h(equation)15414 40940 y Fi(f)142 b Fj(\()p Fi(x)p Fj(\))17948 40392 y Ff(2)18768 40940 y Fe(\241)296 b Fi(f)142 b Fj(\()p Fi(x)p Fj(\))22631 40392 y Ff(3)23525 40940 y Fj(=)369 b Fi(x)25645 40392 y Ff(2)26466 40940 y Fe(\241)295 b Fi(x)28533 40392 y Ff(3)29059 40940 y Fi(;)2823 b Fj(0)369 b Fe(\267)h Fi(x)f Fe(\267)g Fj(1)p Fi(;)12952 b Fj(\(3\))800 43528 y(as)434 b(can)g(easily)h(b)36 b(e)433 b(v)-36 b(eri\257ed.)578 b(A)434 b(simple)f(corollary)j(of)e(this)f(is)h(that)f Fi(f)142 b Fj(\()p Fi(f)g Fj(\()p Fi(x)p Fj(\)\))369 b(=)f Fi(x)434 b Fj(for)g Fi(x)369 b Fe(2)g Fj([0)p Fi(;)221 b Fj(1].)2751 45133 y(Next,)434 b(de\257ne)17089 47676 y Fi(g)48 b Fj(\()p Fi(x)p Fj(\))368 b(=)21748 46016 y Fc(1)21260 46414 y Fb(X)21361 49243 y Fh(k)24 b Ff(=1)24254 46778 y Fj(2\(3)p Fi(k)341 b Fe(\241)295 b Fj(2\)!)p 23532 47371 7113 54 v 23532 48588 a(\(2)p Fi(k)45 b Fj(\)!\()p Fi(k)341 b Fe(\241)295 b Fj(1\)!)30777 47676 y(\()p Fi(x)32022 47128 y Ff(2)32843 47676 y Fe(\241)h Fi(x)34911 47128 y Ff(3)35437 47676 y Fj(\))35943 47128 y Fh(k)800 51111 y Fj(Since)532 b(on)h([0)p Fi(;)221 b Fj(1],)560 b(the)532 b(maximal)i(v)-72 b(alue)534 b(attained)e(b)-36 b(y)533 b Fi(x)28809 50629 y Ff(2)29698 51111 y Fe(\241)362 b Fi(x)31832 50629 y Ff(3)32891 51111 y Fj(is)533 b(4)p Fi(=)p Fj(27)h(\(at)f Fi(x)538 b Fj(=)f(2)p Fi(=)p Fj(3\),)559 b(b)-36 b(y)533 b(Stirling's)800 52716 y(form)-36 b(ula)454 b(the)g(series)g(is)g(seen)g(to)g(con)-36 b(v)g(erge)454 b(ev)-36 b(erywhere)455 b(on)e([0)p Fi(;)221 b Fj(1],)461 b(to)454 b(a)h(function)e Fi(g)48 b Fj(\()p Fi(x)p Fj(\))453 b(whic)-36 b(h)454 b(is)g(real-)800 54321 y(analytic)561 b(except)f(at)g Fi(x)584 b Fj(=)f(2)p Fi(=)p Fj(3.)958 b(W)-108 b(e)560 b(no)-36 b(w)560 b(expand)f Fi(g)48 b Fj(\()p Fi(x)p Fj(\))559 b(in)h(p)36 b(o)-36 b(w)g(ers)559 b(of)i(1)381 b Fe(\241)h Fi(x)p Fj(;)623 b(all)561 b(rearrangemen)-36 b(t)800 55926 y(op)36 b(erations)434 b(are)g(p)36 b(ermitted)432 b(b)-36 b(y)434 b(absolute)f(con)-36 b(v)g(ergence:)16079 59452 y Fi(g)48 b Fj(\()p Fi(x)p Fj(\))369 b(=)20739 57792 y Fc(1)20250 58190 y Fb(X)20352 61019 y Fh(k)24 b Ff(=1)23245 58554 y Fj(2\(3)p Fi(k)341 b Fe(\241)295 b Fj(2\)!)p 22523 59147 V 22523 60363 a(\(2)p Fi(k)45 b Fj(\)!\()p Fi(k)341 b Fe(\241)295 b Fj(1\)!)29768 59452 y Fi(x)30507 58904 y Ff(2)p Fh(k)31547 59452 y Fj(\(1)g Fe(\241)g Fi(x)p Fj(\))35571 58904 y Fh(k)36509 59452 y Fj(=)11673 64653 y(=)13542 62993 y Fc(1)13053 63391 y Fb(X)13155 66220 y Fh(k)24 b Ff(=1)16048 63755 y Fj(2\(3)p Fi(k)341 b Fe(\241)295 b Fj(2\)!)p 15326 64348 V 15326 65565 a(\(2)p Fi(k)45 b Fj(\)!\()p Fi(k)341 b Fe(\241)295 b Fj(1\)!)22571 64653 y(\(1)h Fe(\241)f Fi(x)p Fj(\))26596 64105 y Fh(k)28089 62993 y(k)27386 63391 y Fb(X)27529 66191 y Fh(j)51 b Ff(=0)29526 62780 y Fb(\265)30504 63755 y Fj(2)p Fi(k)30883 65565 y(j)31875 62780 y Fb(\266)32853 64653 y Fj(\()p Fe(\241)p Fj(1\))35548 64105 y Fh(j)36035 64653 y Fj(\(1)296 b Fe(\241)f Fi(x)p Fj(\))40060 64105 y Fh(j)40916 64653 y Fj(=)8461 69901 y(=)10331 68240 y Fc(1)9842 68639 y Fb(X)9915 71429 y Fh(n)p Ff(=1)11982 67231 y Fb(0)11982 69622 y(@)14656 68240 y Fh(n)13982 68639 y Fb(X)13144 71517 y Fh(k)24 b Ff(=)p Fc(d)p Fh(n=)p Ff(3)p Fc(e)16959 68028 y Fb(\265)18812 69002 y Fj(2)p Fi(k)17937 70812 y(n)296 b Fe(\241)f Fi(k)21058 68028 y Fb(\266)22036 69901 y Fj(\()p Fe(\241)p Fj(1\))24731 69352 y Fh(n)p Ff(+)p Fh(k)27457 69002 y Fj(2\(3)p Fi(k)341 b Fe(\241)295 b Fj(2\)!)p 26735 69595 V 26735 70812 a(\(2)p Fi(k)45 b Fj(\)!\()p Fi(k)341 b Fe(\241)295 b Fj(1\)!)33980 67231 y Fb(1)33980 69622 y(A)35364 69901 y Fj(\(1)g Fe(\241)h Fi(x)p Fj(\))39389 69352 y Fh(n)40384 69901 y Fj(=)369 b(1)295 b Fe(\241)g Fi(x;)p Black 26475 74617 a Fj(2)p Black eop %%Page: 3 3 3 2 bop Black 0 TeXcolorgray Black Black 800 1424 a Fj(where)434 b(the)f(last)h(equalit)-36 b(y)434 b(follo)-36 b(ws)436 b(from)e(the)f(automatically)i(v)-36 b(eri\257able)434 b([)p 0 1 0 0 TeXcolorcmyk(5)p (#cite.5) [[413 704 419 716] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])g(iden)-36 b(tit)g(y)14707 3681 y Fh(n)14033 4079 y Fb(X)13196 6957 y Fh(k)24 b Ff(=)p Fc(d)p Fh(n=)p Ff(3)p Fc(e)20006 4443 y Fj(\()p Fe(\241)p Fj(1\))22701 3961 y Fh(k)23270 4443 y Fj(\(3)p Fi(k)340 b Fe(\241)295 b Fj(2\)!)p 17144 5036 14007 54 v 17144 6252 a(\()p Fi(k)340 b Fe(\241)295 b Fj(1\)!\()p Fi(n)h Fe(\241)f Fi(k)45 b Fj(\)!\(3)p Fi(k)341 b Fe(\241)295 b Fi(n)p Fj(\)!)31652 5341 y(=)368 b(0)p Fi(;)2824 b(n)369 b(>)g Fj(1)p Fi(:)800 9718 y Fj(W)-108 b(e)520 b(ha)-36 b(v)g(e)520 b(sho)-36 b(wn)520 b(that)g Fi(g)48 b Fj(\()p Fi(x)p Fj(\))515 b(=)h(1)354 b Fe(\241)h Fi(x)520 b Fj(near)g Fi(x)c Fj(=)g(1.)838 b(But)519 b(since)i Fi(g)48 b Fj(\()p Fi(x)p Fj(\))519 b(is)h(de\257ned)f(as)h(a)h(function)e(of)800 11324 y Fi(x)1539 10842 y Ff(2)2360 11324 y Fe(\241)296 b Fi(x)4428 10842 y Ff(3)4953 11324 y Fj(,)434 b(b)-36 b(y)452 b(\()p 0 1 0 0 TeXcolorcmyk(3)p (#equation.3) [[145 615 151 627] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))433 b(it)h(follo)-36 b(ws)436 b(that)d Fi(g)48 b Fj(\()p Fi(f)142 b Fj(\()p Fi(x)p Fj(\)\))367 b(=)i Fi(g)48 b Fj(\()p Fi(x)p Fj(\),)433 b(and)g(therefore)h(near)f Fi(x)369 b Fj(=)g(0)434 b(w)-36 b(e)434 b(ha)-36 b(v)g(e)12053 15241 y Fi(g)48 b Fj(\()p Fi(x)p Fj(\))369 b(=)f Fi(g)48 b Fj(\()p Fi(f)142 b Fj(\()p Fi(x)p Fj(\)\))368 b(=)h(1)295 b Fe(\241)g Fi(f)142 b Fj(\()p Fi(x)p Fj(\))369 b(=)g Fi(x)29486 14692 y Ff(2)30307 15241 y Fj(+)32103 13580 y Fc(1)31614 13979 y Fb(X)31686 16768 y Fh(n)p Ff(=3)33532 15241 y Fj(\()p Fe(\241)p Fj(1\))36227 14692 y Fh(n)36854 15241 y Fi(m)37992 15440 y Fh(n)p Fc(\241)p Ff(2)39820 15241 y Fi(x)40559 14692 y Fh(n)41185 15241 y Fi(:)800 19308 y Fj(No)-36 b(w)437 b(to)f(pro)-36 b(v)g(e)436 b(\()p 0 1 0 0 TeXcolorcmyk(1)p (#equation.1) [[156 543 162 555] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))o(,)h(w)-36 b(e)436 b(expand)g Fi(g)48 b Fj(\()p Fi(x)p Fj(\))435 b(in)-36 b(to)436 b(p)36 b(o)-36 b(w)g(ers)436 b(of)h Fi(x)p Fj(,)f(again)h(using)f(easily)h(justi\257able)f (rearrange-)800 20913 y(men)-36 b(t)433 b(op)36 b(erations)16079 23457 y Fi(g)48 b Fj(\()p Fi(x)p Fj(\))369 b(=)20739 21796 y Fc(1)20250 22195 y Fb(X)20352 25024 y Fh(k)24 b Ff(=1)23245 22558 y Fj(2\(3)p Fi(k)341 b Fe(\241)295 b Fj(2\)!)p 22523 23151 7113 54 v 22523 24368 a(\(2)p Fi(k)45 b Fj(\)!\()p Fi(k)341 b Fe(\241)295 b Fj(1\)!)29768 23457 y Fi(x)30507 22908 y Ff(2)p Fh(k)31547 23457 y Fj(\(1)g Fe(\241)g Fi(x)p Fj(\))35571 22908 y Fh(k)36509 23457 y Fj(=)15048 28383 y(=)16918 26722 y Fc(1)16429 27121 y Fb(X)16530 29950 y Fh(k)24 b Ff(=1)19423 27484 y Fj(2\(3)p Fi(k)341 b Fe(\241)296 b Fj(2\)!)p 18702 28077 V 18702 29294 a(\(2)p Fi(k)45 b Fj(\)!\()p Fi(k)340 b Fe(\241)295 b Fj(1\)!)25947 28383 y Fi(x)26686 27834 y Ff(2)p Fh(k)28649 26722 y(k)27947 27121 y Fb(X)28089 29920 y Fh(j)51 b Ff(=0)30087 26510 y Fb(\265)31064 27484 y Fi(k)31119 29294 y(j)31786 26510 y Fb(\266)32763 28383 y Fj(\()p Fe(\241)p Fj(1\))35458 27834 y Fh(j)35945 28383 y Fi(x)36684 27834 y Fh(j)37540 28383 y Fj(=)11438 33766 y(=)13308 32106 y Fc(1)12819 32504 y Fb(X)12892 35294 y Fh(n)p Ff(=2)14959 31096 y Fb(0)14959 33487 y(@)16121 33766 y Fj(\()p Fe(\241)p Fj(1\))18816 33218 y Fh(n)20287 32003 y Fc(b)p Fh(n=)p Ff(2)p Fc(c)20502 32504 y Fb(X)19664 35382 y Fh(k)24 b Ff(=)p Fc(d)p Fh(n=)p Ff(3)p Fc(e)28756 32868 y Fj(\(3)p Fi(k)341 b Fe(\241)295 b Fj(2\)!)p 23612 33461 15307 54 v 23612 34677 a(\(2)p Fi(k)340 b Fe(\241)296 b Fj(1\)!\()p Fi(n)f Fe(\241)h Fj(2)p Fi(k)45 b Fj(\)!\(3)p Fi(k)341 b Fe(\241)295 b Fi(n)p Fj(\)!)39051 31096 y Fb(1)39051 33487 y(A)40435 33766 y Fi(x)41174 33218 y Fh(n)41800 33766 y Fi(:)800 37639 y Fj(Equating)434 b(co)36 b(e\261cien)-36 b(ts)434 b(in)f(the)g(last)h(t)-36 b(w)g(o)434 b(form)-36 b(ulas)434 b(giv)-36 b(es)435 b(\()p 0 1 0 0 TeXcolorcmyk(1)p (#equation.1) [[348 378 354 390] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\).)p 52136 37639 665 665 v 800 41755 a Fd(Pro)42 b(of)700 b(of)836 b Fj(\()p 0 1 0 0 TeXcolorcmyk(2)p (#equation.2) [[141 341 147 353] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))p Fd(.)1301 b Fj(W)-108 b(e)609 b(use)h(a)f(similar)i(idea,) 654 b(this)609 b(time)h(using)f(instead)g(of)h(the)f(function)g Fi(f)142 b Fj(\()p Fi(x)p Fj(\))610 b(the)800 43360 y(function)416 b Fe(\241)221 b Fj(log)241 b Fi(f)142 b Fj(\()p Fi(x)p Fj(\),)419 b(whic)-36 b(h)416 b(generates)g(a)g(sequence)g(related)g (to)g Fi(c)34638 43559 y Fh(n)35264 43360 y Fj(.)572 b(Since)416 b(the)f(generating)h(function)800 44965 y(for)434 b Fi(c)3347 45164 y Fh(n)4407 44965 y Fj(is)g(w)-36 b(ell)434 b(kno)-36 b(wn)434 b([)p 0 1 0 0 TeXcolorcmyk(6)p (#cite.6) [[187 312 193 324] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])f(to)h(b)36 b(e)433 b(1)p Fi(=)18937 43873 y Fe(p)p 20045 43873 7203 54 v 20045 44965 a Fj(1)296 b Fe(\241)f Fj(2)p Fi(x)h Fe(\241)f Fj(3)p Fi(x)26721 44582 y Ff(2)27247 44965 y Fj(,)434 b(it)g(is)g(easy)g(to)g(v)-36 b(erify)435 b(that)18409 47984 y Fi(f)19192 47502 y Fc(0)19502 47984 y Fj(\()p Fi(x)p Fj(\))p 18409 48577 2845 54 v 18564 49794 a Fi(f)142 b Fj(\()p Fi(x)p Fj(\))21755 48883 y(=)23625 47222 y Fc(1)23136 47621 y Fb(X)23208 50410 y Fh(n)p Ff(=0)25409 47984 y Fj(\()p Fe(\241)p Fj(1\))28104 47502 y Fh(n)28730 47984 y Fi(c)29290 48183 y Fh(n)p Ff(+1)31414 47984 y Fe(\241)295 b Fj(1)p 25409 48577 7984 54 v 29075 49794 a(2)33959 48883 y Fi(x)34698 48334 y Fh(n)800 52876 y Fj(and)433 b(therefore)17418 55161 y Fe(\241)221 b Fj(log)241 b Fi(f)142 b Fj(\()p Fi(x)p Fj(\))368 b(=)25346 53501 y Fc(1)24857 53899 y Fb(X)24930 56689 y Fh(n)p Ff(=1)27130 54263 y Fj(\()p Fe(\241)p Fj(1\))29825 53781 y Fh(n)30451 54263 y Fi(c)31011 54462 y Fh(n)31932 54263 y Fj(+)295 b(1)p 27130 54856 6760 54 v 29796 56072 a(2)p Fi(n)34456 55161 y(x)35195 54613 y Fh(n)35821 55161 y Fi(:)800 58562 y Fj(No)-36 b(w)434 b(de\257ne)f(the)g(function)18097 60847 y Fi(h)p Fj(\()p Fi(x)p Fj(\))368 b(=)22835 59187 y Fc(1)22346 59585 y Fb(X)22447 62414 y Fh(k)24 b Ff(=1)24619 59949 y Fj(\(3)p Fi(k)340 b Fe(\241)295 b Fj(1\)!)p 24619 60542 5019 54 v 25214 61759 a Fi(k)45 b Fj(!\(2)p Fi(k)g Fj(\)!)29770 60847 y(\()p Fi(x)31015 60299 y Ff(2)31836 60847 y Fe(\241)295 b Fi(x)33903 60299 y Ff(3)34429 60847 y Fj(\))34935 60299 y Fh(k)800 64361 y Fj(whic)-36 b(h)509 b(again)h(con)-36 b(v)g(erges)509 b(for)h(all)g Fi(x)497 b Fe(2)g Fj([0)p Fi(;)221 b Fj(1])511 b(to)e(a)h(function)f(whic)-36 b(h)508 b(is)i(analytic)g(except)f(at)g Fi(x)498 b Fj(=)f(2)p Fi(=)p Fj(3.)800 65967 y(Expanding)433 b Fi(h)p Fj(\()p Fi(x)p Fj(\))g(in)-36 b(to)434 b(p)36 b(o)-36 b(w)g(ers)433 b(of)i(1)295 b Fe(\241)h Fi(x)433 b Fj(giv)-36 b(es)11285 70117 y Fi(h)p Fj(\()p Fi(x)p Fj(\))369 b(=)16023 68457 y Fc(1)15535 68855 y Fb(X)15636 71684 y Fh(k)24 b Ff(=1)17807 69219 y Fj(\(3)p Fi(k)341 b Fe(\241)295 b Fj(1\)!)p 17807 69812 V 18403 71029 a Fi(k)45 b Fj(!\(2)p Fi(k)g Fj(\)!)22958 70117 y(\(1)296 b Fe(\241)f Fi(x)p Fj(\))26983 69569 y Fh(k)28241 68457 y Ff(2)p Fh(k)27773 68855 y Fb(X)27916 71655 y Fh(j)51 b Ff(=0)29913 68244 y Fb(\265)30891 69219 y Fj(2)p Fi(k)31271 71029 y(j)32263 68244 y Fb(\266)33240 70117 y Fj(\()p Fe(\241)p Fj(1\))35935 69569 y Fh(j)36423 70117 y Fj(\(1)295 b Fe(\241)g Fi(x)p Fj(\))40447 69569 y Fh(j)41303 70117 y Fj(=)p Black 26475 74617 a(3)p Black eop %%Page: 4 4 4 3 bop Black 0 TeXcolorgray Black Black 11380 2819 a Fj(=)13249 1158 y Fc(1)12760 1557 y Fb(X)12833 4347 y Fh(n)p Ff(=1)14901 149 y Fb(0)14901 2540 y(@)17574 1158 y Fh(n)16900 1557 y Fb(X)16063 4435 y Fh(k)24 b Ff(=)p Fc(d)p Fh(n=)p Ff(3)p Fc(e)19878 946 y Fb(\265)21730 1920 y Fj(2)p Fi(k)20856 3730 y(n)295 b Fe(\241)h Fi(k)23977 946 y Fb(\266)24954 2819 y Fj(\()p Fe(\241)p Fj(1\))27649 2270 y Fh(n)p Fc(\241)p Fh(k)29654 1920 y Fj(\(3)p Fi(k)340 b Fe(\241)296 b Fj(1\)!)p 29654 2513 5019 54 v 30250 3730 a Fi(k)45 b Fj(!\(2)p Fi(k)g Fj(\)!)34805 149 y Fb(1)34805 2540 y(A)36189 2819 y Fj(\(1)295 b Fe(\241)g Fi(x)p Fj(\))40213 2270 y Fh(n)41209 2819 y Fj(=)19578 7883 y(=)21448 6222 y Fc(1)20959 6621 y Fb(X)21032 9410 y Fh(n)p Ff(=1)23232 6984 y Fj(\(1)g Fe(\241)h Fi(x)p Fj(\))27257 6502 y Fh(n)p 23232 7577 4651 54 v 25169 8794 a Fi(n)28385 7883 y Fj(=)368 b Fe(\241)221 b Fj(log)241 b Fi(x;)800 11370 y Fj(again)434 b(making)h(use)e(of)i(a)e(v)-36 b(eri\257able)434 b(iden)-36 b(tit)g(y)434 b([)p 0 1 0 0 TeXcolorcmyk(5)p (#cite.5) [[292 615 297 627] [1 1 1 [3 3]] [0 0 1]] pdfm Black(],)g(namely)h(that)12861 15315 y(\()p Fe(\241)p Fj(1\))15556 14767 y Fh(n)17915 13655 y(n)17241 14053 y Fb(X)16403 16931 y Fh(k)24 b Ff(=)p Fc(d)p Fh(n=)p Ff(3)p Fc(e)21570 14417 y Fj(\()p Fe(\241)p Fj(1\))24265 13935 y Fh(k)24834 14417 y Fj(\(3)p Fi(k)341 b Fe(\241)295 b Fj(1\)!)p 20351 15010 10721 54 v 20351 16226 a Fi(k)45 b Fj(!\()p Fi(n)296 b Fe(\241)f Fi(k)45 b Fj(\)!\(3)p Fi(k)341 b Fe(\241)295 b Fi(n)p Fj(\)!)31574 15315 y(=)33150 14417 y(1)p 33087 15010 777 54 v 33087 16226 a Fi(n)33996 15315 y(;)2823 b(n)370 b Fe(\270)f Fj(1)p Fi(:)10399 b Fj(\(4\))800 19721 y(So)448 b Fi(h)p Fj(\()p Fi(x)p Fj(\))393 b(=)h Fe(\241)221 b Fj(log)241 b Fi(x)448 b Fj(near)g Fi(x)393 b Fj(=)h(1,)452 b(and)c(therefore)g(b)36 b(ecause)448 b(of)h(the)e(symmetry)i(prop)36 b(ert)-36 b(y)465 b(\()p 0 1 0 0 TeXcolorcmyk(3)p (#equation.3) [[492 540 498 552] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))448 b(w)-36 b(e)449 b(ha)-36 b(v)g(e)800 21326 y(that)433 b Fi(h)p Fj(\()p Fi(x)p Fj(\))369 b(=)f Fe(\241)221 b Fj(log)241 b Fi(f)142 b Fj(\()p Fi(x)p Fj(\))433 b(near)h Fi(x)369 b Fj(=)g(0.)578 b(Expanding)433 b Fi(h)p Fj(\()p Fi(x)p Fj(\))h(in)f(p)36 b(o)-36 b(w)g(ers)434 b(of)g Fi(x)g Fj(near)f Fi(x)369 b Fj(=)g(0)434 b(giv)-36 b(es)10941 25505 y Fe(\241)221 b Fj(log)241 b Fi(f)142 b Fj(\()p Fi(x)p Fj(\))369 b(=)f Fi(h)p Fj(\()p Fi(x)p Fj(\))h(=)23118 23845 y Fc(1)22630 24244 y Fb(X)22731 27072 y Fh(k)24 b Ff(=1)24903 24607 y Fj(\(3)p Fi(k)340 b Fe(\241)295 b Fj(1\)!)p 24903 25200 5019 54 v 25498 26417 a Fi(k)45 b Fj(!\(2)p Fi(k)g Fj(\)!)30054 25505 y Fi(x)30793 24957 y Ff(2)p Fh(k)32756 23845 y(k)32053 24244 y Fb(X)32196 27043 y Fh(j)51 b Ff(=0)34193 23632 y Fb(\265)35171 24607 y Fi(k)35225 26417 y(j)35892 23632 y Fb(\266)36870 25505 y Fj(\()p Fe(\241)p Fj(1\))39565 24957 y Fh(j)40052 25505 y Fi(x)40791 24957 y Fh(j)41647 25505 y Fj(=)12756 31510 y(=)14625 29850 y Fc(1)14136 30248 y Fb(X)14209 33038 y Fh(n)p Ff(=2)16277 28840 y Fb(0)16277 31231 y(@)17439 31510 y Fj(\()p Fe(\241)p Fj(1\))20134 30962 y Fh(n)21604 29746 y Fc(b)p Fh(n=)p Ff(2)p Fc(c)21819 30248 y Fb(X)20982 33126 y Fh(k)24 b Ff(=)p Fc(d)p Fh(n=)p Ff(3)p Fc(e)28937 30611 y Fj(\(3)p Fi(k)340 b Fe(\241)296 b Fj(1\)!)p 24929 31205 13033 54 v 24929 32421 a(\(2)p Fi(k)45 b Fj(\)!\()p Fi(n)296 b Fe(\241)g Fj(2)p Fi(k)45 b Fj(\)!\(3)p Fi(k)340 b Fe(\241)296 b Fi(n)p Fj(\)!)38095 28840 y Fb(1)38095 31231 y(A)39479 31510 y Fi(x)40218 30962 y Fh(n)800 35396 y Fj(Equating)434 b(co)36 b(e\261cien)-36 b(ts)434 b(with)f(our)h (previous)f(expansion)h(of)h Fi(h)p Fj(\()p Fi(x)p Fj(\))e(giv)-36 b(es)434 b(\()p 0 1 0 0 TeXcolorcmyk(2)p (#equation.2) [[410 398 416 410] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\).)p 52136 35396 665 665 v 800 39524 a Fd(Remarks.)p Black 2390 42236 a Fj(1.)p Black 651 w(One)415 b(ob)-36 b(vious)416 b(question)g(on)g(seeing)h(form)-36 b(ulas)416 b(\()p 0 1 0 0 TeXcolorcmyk(1)p (#equation.1) [[324 337 330 349] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))g(and)f(\()p 0 1 0 0 TeXcolorcmyk(2)p (#equation.2) [[365 337 371 349] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))h(is,)k(Can)d(they)f(b)36 b(e)415 b(explained)h(com)-36 b(bi-)4052 43841 y(natorially?)571 b(That)408 b(is,)413 b(do)407 b(there)g(exist)i(bijections)f(b)36 b(et)-36 b(w)g(een)407 b(sets)g(kno)-36 b(wn)408 b(to)g(b)36 b(e)407 b(en)-36 b(umerated)406 b(b)-36 b(y)4052 45446 y(the)454 b(n)-36 b(um)g(b)36 b(ers)453 b Fi(m)12718 45645 y Fh(n)13799 45446 y Fj(and)i Fi(c)16910 45645 y Fh(n)17536 45446 y Fj(,)460 b(and)455 b(sets)g(whose)g(cardinalit)-36 b(y)456 b(is)f(seen)g(to)g(b)36 b(e)454 b(the)h(righ)-36 b(t-hand)453 b(sides)4052 47051 y(of)527 b(\()p 0 1 0 0 TeXcolorcmyk(1)p (#equation.1) [[127 294 133 306] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))433 b(and)g(\()p 0 1 0 0 TeXcolorcmyk(2)p (#equation.2) [[169 294 175 306] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\)?)578 b(Suc)-36 b(h)433 b(explanations)h(elude)f(us)g(curren) -36 b(tly)-108 b(.)p Black 2390 49763 a(2.)p Black 651 w(Iden)-36 b(tit)g(y)417 b(\()p 0 1 0 0 TeXcolorcmyk(4)p (#equation.4) [[157 269 163 281] [1 1 1 [3 3]] [0 0 1]] pdfm Black(\))400 b(is)g(a)h(sp)36 b(ecial)400 b(case)h(of)f(a)h(more)f (general)g(iden)-36 b(tit)g(y)400 b([)p 0 1 0 0 TeXcolorcmyk(4)p (#cite.4) [[392 269 398 281] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)407 b(Eq.)401 b(\(6\)])f(that)f(w)-36 b(as)401 b(disco)-36 b(v)g(ered)4052 51368 y(b)g(y)433 b(Thomas)h(Liggett.)p Black 2390 54080 a(3.)p Black 651 w(See)624 b([)p 0 1 0 0 TeXcolorcmyk(1)p (#cite.1) [[134 230 140 242] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)p 0 1 0 0 TeXcolorcmyk 625 w(2)p (#cite.2) [[149 230 155 242] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)p 0 1 0 0 TeXcolorcmyk 625 w(3)p (#cite.3) [[164 230 170 242] [1 1 1 [3 3]] [0 0 1]] pdfm Black(,)p 0 1 0 0 TeXcolorcmyk 625 w(6)p (#cite.6) [[178 230 184 242] [1 1 1 [3 3]] [0 0 1]] pdfm Black 1 w(])g(for)h(some)g(other)f(form)-36 b(ulas)625 b(in)-36 b(v)g(olving)626 b(the)d(cen)-36 b(tral)625 b(trinomial)g(co)36 b(e\261cien)-36 b(ts)4052 55685 y(and)435 b(the)h(Motzkin)g(n)-36 b(um)g(b)36 b(ers,)435 b(and)g(for)i(more)f (information)h(on)e(the)h(prop)36 b(erties,)436 b(and)g(the)f(man)-36 b(y)4052 57290 y(di\256eren)g(t)432 b(com)-36 b(binatorial)435 b(in)-36 b(terpretations,)433 b(of)h(these)f(sequences.)800 61727 y Fk(3)2152 b(Ac)-60 b(kno)g(wledgmen)g(ts)800 64648 y Fj(Thanks)434 b(to)g(the)f(anon)-36 b(ymous)433 b(referee)h(for)g(some)g(useful)g(suggestions)g(and)f(references.)p Black 26475 74617 a(4)p Black eop %%Page: 5 5 5 4 bop Black 0 TeXcolorgray Black Black 800 1424 a Fk(References)p Black 800 4345 a Fj([1])p Black 651 w(M.)434 b(Aigner,)g(Motzkin)g(n) -36 b(um)g(b)36 b(ers.)432 b Fg(Eur)-66 b(op)g(e)g(an)464 b(J.)h(Combin.)432 b Fj(19)i(\(1998\),)h(663{675.)p Black 800 7057 a([2])p Black 651 w(E.)397 b(Barcucci,)405 b(R.)397 b(Pinzani)f(and)g(R.)h(Sprugnoli,)404 b(The)396 b(Motzkin)h(family)-108 b(.)398 b Fg(Pur)-66 b(e)431 b(Math.)f(Appl.)h(Ser.)f(A)2823 8662 y Fj(2)k(\(1991\),)h(249{279.)p Black 800 11374 a([3])p Black 651 w(R.)355 b(Donaghey)g(and)f(L.)g(W.)g(Shapiro,)370 b(Motzkin)354 b(n)-36 b(um)g(b)36 b(ers.)353 b Fg(J.)392 b(Combin.)e(The)-66 b(ory)391 b(Ser.)g(A)353 b Fj(23)i(\(1977\),)2823 12979 y(291{301.)p Black 800 15691 a([4])p Black 651 w(A.)405 b(E.)f(Holro)-36 b(yd,)412 b(D.)404 b(Romik)i(and)e(T.)g(M.)h (Liggett,)411 b(In)-36 b(tegrals,)410 b(partitions)405 b(and)e(cellular)i(automata.)2823 17296 y(T)-108 b(o)434 b(app)36 b(ear)434 b(in)f Fg(T)-100 b(r)-66 b(ans.)464 b(A)-33 b(mer.)464 b(Math.)g(So)-66 b(c.)p Black 800 20008 a Fj([5])p Black 651 w(M.)434 b(P)-36 b(etk)g(o)g(v)-69 b(\267)-582 b(sek,)436 b(H.)e(S.)f(Wilf)i(and)e(D.)i(Zeilb)36 b(erger,)434 b Fi(A)369 b Fj(=)f Fi(B)67 b Fj(,)434 b(A.)g(K.)g(P)-36 b(eters,)434 b(1996.)p Black 800 22720 a([6])p Black 651 w(N.)841 b(J.)h(A.)f(Sloane,)943 b(editor)841 b(\(2003\),)943 b(The)841 b(On-Line)e(Encyclop)36 b(edia)842 b(of)g(In)-36 b(teger)840 b(Sequences,)2823 24325 y Fa(http://www.research.att.com/) 17 b Fj(~)t Fa(njas/sequences/)p Fj(,)427 b(sequences)434 b Fa(A002426)p Fj(,)d Fa(A001006)p Fj(.)p 800 27432 52000 45 v 800 29683 a(2000)k Fg(Mathematics)464 b(Subje)-66 b(ct)463 b(Classi\257c)-66 b(ation)p Fj(:)577 b(05A10,)435 b(05A15,)g(05A19.)800 31288 y Fg(Keywor)-66 b(ds:)1250 b Fj(cen)-36 b(tral)434 b(trinomial)g(co)36 b(e\261cien)-36 b(ts,)434 b(Motzkin)g(n)-36 b(um)g(b)36 b(ers,)432 b(binomial)i(iden) -36 b(tities.)p 800 32919 V 800 35244 a(\(Concerned)433 b(with)h(sequences)p 0 1 0 0 TeXcolorcmyk 433 w(A001006)p 16603 35456 4878 54 v [[221 400 265 412] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A001006) pdfm Black 436 w(and)p 0 1 0 0 TeXcolorcmyk 433 w(A002426)p 24443 35456 V [[292 400 336 412] [1 1 1 [3 3]] [0 0 1]] (http://www.research.att.com/cgi-bin/access.cgi/as/~njas/sequences/eisA.cgi?Anum=A002426) pdfm Black 2 w(.\))800 37649 y(Receiv)-36 b(ed)503 b(Marc)-36 b(h)502 b(25,)521 b(2003;)539 b(revised)503 b(v)-36 b(ersion)503 b(receiv)-36 b(ed)502 b(June)g(20,)521 b(2003.)787 b(Published)501 b(in)i Fg(Journal)528 b(of)800 39254 y(Inte)-66 b(ger)463 b(Se)-66 b(quenc)g(es)431 b Fj(July)j(8,)g(2003.)p 800 40885 52000 45 v 800 43136 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 [[133 329 338 341] [1 1 1 [3 3]] [0 0 1]] (http://www.math.uwaterlo o.ca/JIS/) pdfm Black(.)p Black 26475 74617 a(5)p Black eop %%Trailer end end