Supercongruences for central trinomial coefficients

2020/12/09 by Pan, Hao, Sun, Zhi-Wei
#Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2012.05121

Abstract

For each n=0,1,2,… the central trinomial coefficient T_n is the coefficient of xⁿ in the expansion of (x²+x+1)ⁿ. In 2016 the second author conjectured that for any prime p>3 and positive integer n the quotient (T_pn-T_n)/(pn)² is a p-adic integer. In this paper we confirm this conjecture and prove further that \fracT_pn-T_n(pn)²≡\fracT_n-16(\frac p3)B_p-2(\frac13)\pmod p, where (\frac p3) is the Legendre symbol and B_p-2(x) is the Bernoulli polynomial of degree p-2.

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