Journal of Integer Sequences, Vol. 6 (2003), Article 03.3.3

Matrix Transformations of Integer Sequences


Clark Kimberling
Department of Mathematics
University of Evansville
1800 Lincoln Avenue
Evansville, IN 47722

Abstract: The integer sequences with first term $ 1$ comprise a group $ \mathcal{G}$ under convolution, namely, the Appell group, and the lower triangular infinite integer matrices with all diagonal entries $ 1$ comprise a group $ \mathbb{G}$ under matrix multiplication. If $ A\in \mathcal{G}$ and $ M\in \mathbb{G},$ then $ MA\in \mathcal{G}.$ The groups $ %% \mathcal{G}$ and $ \mathbb{G}$ and various subgroups are discussed. These include the group $ \mathbb{G}^{(1)}$ of matrices whose columns are identical except for initial zeros, and also the group $ \mathbb{G}^{(2)}$ of matrices in which the odd-numbered columns are identical except for initial zeros and the same is true for even-numbered columns. Conditions are determined for the product of two matrices in $ \mathbb{G}^{(m)}$ to be in $ \mathbb{G}%% ^{(1)}. $ Conditions are also determined for two matrices in $ \mathbb{G}%% ^{(2)}$ to commute.


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(Concerned with sequences A000045 A000108 A000142 A000201 A000204 A000741 A000984 A002530 A047749 A077049 A077050 A077605 A077606 .)

Received November 13, 2002; revised version received January 28, 2002; September 2, 2003. Published in Journal of Integer Sequences September 8, 2003.

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