Congruences involving generalized central trinomial coefficients

Details Congruences involving generalized central trinomial coefficients Congruences involving generalized central trinomial coefficients

2010-08-23

arxiv.org/abs/1008.3887v11

Mathematics

Number Theory

34 pages. (1.24) in Th. 1.4 is new

Scientific paper

For integers $b$ and $c$ the generalized trinomial coefficient $T_n(b,c)$ denotes the coefficient of $x^n$ in the expansion of $(x^2+bx+c)^n$. Those $T_n=T_n(1,1) (n=0,1,2,...)$ are the usual central trinomial coefficients, and $T_n(3,2)$ coincides with the Delannoy number $D_n=\sum_{k=0}^n\binom(n,k)\binom(n+k,k)$. In this paper we investigate congruences involving generalized central trinomial coefficients systematically. Here are some typical results: For each $n=1,2,3,...$ we have $$\sum_{k=0}^{n-1}(2k+1)T_k(b,c)^2(b^2-4c)^{n-1-k}=0 (mod n^2}$$ and in particular $n^2|\sum_{k=0}^{n-1}(2k+1)D_k^2$; if $p$ is an odd prime then $$\sum_{k=0}^{p-1}T_k^2=(-1/p) (mod p) and \sum_{k=0}^{p-1}D_k^2=(2/p) (mod p).$$ We also raise several conjectures some of which involve parameters in the representations of primes by certain binary quadratic forms.

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