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On a combinatorial problem of Asmus Schmidt

Wadim Zudilin
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Abstract

For any integer $r\ge2$, define a sequence of numbers $\{c_k^{(r)}\}_{k=0}^\infty$, independent of the parameter $n$, by $$ \sum_{k=0}^n{\binom nk}^r{\binom{n+k}k}^r =\sum_{k=0}^n\binom nk\binom{n+k}kc_k^{(r)}, \qquad n=0,1,2,...c. $$ We prove that all the numbers $c_k^{(r)}$ are integers.

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Wadim Zudilin. 2003-11-12. On a combinatorial problem of Asmus Schmidt. https://arxiv.org/abs/math/0311195

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