Congruences involving generalized central trinomial coefficients

Abstract

For integers $b$ and $c$ the generalized central trinomial coefficient $T_n(b,c)$ denotes the coefficient of $x^n$ in the expansion of $(x^2+bx+c{)}^n$. Those $T_n=T_n(1,1)(n=0,1,2,\dots )$ are the usual central trinomial coefficients, and $T_n(3,2)$ coincides with the Delannoy number $D_n={\sum }_{k=0}^n\left(\genfrac{}{}{0ex}{}{n}{k}\right)\left(\genfrac{}{}{0ex}{}{n+k}{k}\right)$ in combinatorics. We investigate congruences involving generalized central trinomial coefficients systematically. Here are some typical results: For each $n=1,2,3,\dots$ we have $$\sum _{k=0}^{n-1}(2k+1)T_k(b,c{)}^2(b^2-4c{)}^{n-1-k}\equiv 0(\mathrm{mod}{n}^2)$$ and in particular $n^2\mid {\sum }_{k=0}^{n-1}(2k+1)D_k^2$; if $p$ is an odd prime then $$\sum _{k=0}^{p-1}{T}_k^2\equiv \left(\frac{-1}{p}\right)(\mathrm{mod}p)\mathrm{and}\sum _{k=0}^{p-1}{D}_k^2\equiv \left(\frac{2}{p}\right)(\mathrm{mod}p),$$ where $(-)$ denotes the Legendre symbol. We also raise several conjectures some of which involve parameters in the representations of primes by certain binary quadratic forms.

34 pages

References in corpus (2)

• On Delannoy numbers and Schröder numbers
• Some congruences for trinomial coefficients

Cited by in corpus (3)

• Proof of some conjectures of Z.-W. Sun on congruences for Apery polynomials
• Conjectures and results on $x^2$ mod $p^2$ with $4p=x^2+d{y}^2$
• On divisibility of sums of Apery polynomials