On some divisibility properties of binomial sums

2016/11/04 by Sun, Brian Y.
#11A07 #11B65 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Primary 11A05 #Secondary 05A10

paper · doi:10.48550/arxiv.1611.01358

Abstract

In this paper, we consider two particular binomial sums ∑_k=0^n-1(20k²+8k+1)\binom2kk⁵ (-4096)^n-k-1 and ∑_k=0^n-1(120k²+34k+3)\binom2kk⁴\binom4k2k 65536^n-k-1, which are inspired by two series for (1)/(π²) obtained by Guillera. We consider their divisibility properties and prove that they are divisible by 2n² \binom2nn² for all integer n≥ 2. These divisibility properties are stronger than those divisibility results found by He, who proved the above two sums are divisible by 2n \binom2nn with the WZ-method.

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