Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.7

Deformations of the Taylor Formula

Emmanuel Ferrand
Institut Mathématique de Jussieu
UMR 7586 du CNRS
Université Pierre et Marie Curie
4, place Jussieu
75252 Paris Cedex


Given a sequence $ x=\{x_n, \ n \in \mathbb{N}\}$ with integer values, or more generally with values in a ring of polynomials with integer coefficients, one can form the generalized binomial coefficients associated with $ x$, $ {\binom nm}_x=\prod_{l=1}^{m} \frac{x_{n-l+1}}{x_l}$. In this note we introduce several sequences that possess the following remarkable feature: the fractions $ \binom nm_x$ are in fact polynomials with integer coefficients.

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(Concerned with sequences A000045 A000120 A000215 A000225 and A000317 .)

Received September 22 2005; revised version received October 8 2006. Published in Journal of Integer Sequences, December 31 2006.

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