410 research outputs found
A matrix generalization of a theorem of Fine
In 1947 Nathan Fine gave a beautiful product for the number of binomial
coefficients , for in the range , that are
not divisible by . We give a matrix product that generalizes Fine's formula,
simultaneously counting binomial coefficients with -adic valuation
for each . For each this information is naturally encoded in
a polynomial generating function, and the sequence of these polynomials is
-regular in the sense of Allouche and Shallit. We also give a further
generalization to multinomial coefficients.Comment: 9 pages; publication versio
A Case Study in Meta-AUTOMATION: AUTOMATIC Generation of Congruence AUTOMATA For Combinatorial Sequences
This article is a sequel to a recent article by Eric Rowland and Reem
Yassawi, presenting yet another approach to the fast determination of
congruence properties of `famous' combinatorial sequences. The present approach
can be taught to a computer, and our beloved servant, Shalosh B. Ekhad, was
able to generate many new theorems, for famous sequences, of course, but also
for many obscure ones!Comment: 17 pages, accompanied by Maple package
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