Factors of sums and alternating sums involving binomial coefficients and powers of integers

Abstract

We study divisibility properties of certain sums and alternating sums involving binomial coefficients and powers of integers. For example, we prove that for all positive integers $n_1,...,n_m$, $n_{m+1}=n_1$, and any nonnegative integer $r$, there holds {align*} \sum_{k=0}^{n_1}ε^k (2k+1)^{2r+1}\prod_{i=1}^{m} {n_i+n_{i+1}+1\choose n_i-k} \equiv 0 \mod (n_1+n_m+1){n_1+n_m\choose n_1}, {align*} and conjecture that for any nonnegative integer $r$ and positive integer $s$ such that $r+s$ is odd, $$\left(\genfrac{}{}{0ex}{}{{\sum }_{k=0}^n{\epsilon }^k(2k+1{)}^r(\left(\genfrac{}{}{0ex}{}{2n}{n-k}\right)-\left(\genfrac{}{}{0ex}{}{2n}{n-k-1}\right){)}^s\equiv 0\mathrm{mod}2n}{n,}\right)$$ where $\epsilon =\pm 1$.

14 pages, to appear in Int. J. Number Theory

References in corpus (3)

• A q-rious positivity
• Factors of binomial sums from the Catalan triangle
• Some congruences for the second-order Catalan numbers

Cited by in corpus (2)

• Factors of sums and alternating sums of products of $q$-binomial coefficients and powers of $q$-integers
• Proof of a conjecture related to divisibility properties of binomial coefficients