Proof of some conjectures of Z.-W. Sun on the divisibility of certain double-sums

2014/12/10 by Victor J. W. Guo, Guo, Victor J. W., Ji-Cai Liu +1
Mathematics · #05A10 #11A07 #11B65 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT #msc:05A10 #msc:11A07 #msc:11B65

paper · pdf · doi:10.48550/arxiv.1412.5415

8 pages

arxiv created 2014/12/10 · arxiv updated 2014/12/18

Abstract

Z.-W. Sun introduced three kinds of numbers: S_n=∑_k=0ⁿn\choose k²2k\choose k(2k+1), s_n=∑_k=0ⁿn\choose k²2k\choose k(1)/(2k-1), and S_n⁺=∑_k=0ⁿn\choose k²2k\choose k(2k+1)². In this paper we mainly prove that 4∑_k=0^n-1kS_k≡ ∑_k=0^n-1s_k≡ ∑_k=0^n-1S_k⁺≡ 0\pmodn²\quadfor n\geqslant 1, by establishing some binomial coefficient identities, such as 4∑_k=0^n-1kS_k=n²∑_k=0^n-1(1)/(k+1)2k\choose k(6kn-1\choose k²+n-1\choose kn-1\choose k+1). This confirms several recent conjectures of Z.-W. Sun.

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