On a combinatorial problem of Asmus Schmidt. (English) Zbl 1126.11012

Zudilin, W.

Electron. J. Comb. 11, No. 1, Research paper R22, 8 p. (2004).

Summary: For any integer \(r\geq 2\), define a sequence of numbers \(\{c^{(r)}_k\}_{k=0,1,\dots}\), independent of the parameter \(n\), by \[ \sum^n_{k=0}\binom nk^r\binom{n+k}{k}^r=\sum^n_{k=0}\binom nk\binom{n+k}{k}c^{(r)}_k,\quad n=0,1,2,\dots. \] We prove that all the numbers \(c^{(r)}_k\) are integers.

MSC:

11B65	Binomial coefficients; factorials; \(q\)-identities
05A19	Combinatorial identities, bijective combinatorics