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The p-adic valuation of k-central binomial coefficients
Armin Straub, Victor H. Moll, Tewodros Amdeberhan — Acta Arithmetica — Volume 140, Number 1, 2009, Pages 31-42

Abstract

The coefficients

c(n,k)

defined by

(1-k^{2}x)^{-1/k} = \sum_{n \geq 0} c(n,k)x^n

reduce to the central binomial coefficients

\binom{2n}{n}

for

k=2

. Motivated by a question of H. Montgomery and H. Shapiro for the case

k=3

, we prove that

c(n,k)

are integers and study their divisibility properties.

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BibTeX

@article{centralbinomials-2009,
    author = {Armin Straub and Victor H. Moll and Tewodros Amdeberhan},
    title = {The $p$-adic valuation of $k$-central binomial coefficients},
    journal = {Acta Arithmetica},
    year = {2009},
    volume = {140},
    number = {1},
    pages = {31--42},
    doi = {10.4064/aa140-1-2},
}