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Abstract: For integers $b$ and $c$ the generalized central trinomial coefficient$T nb,c$ denotes the coefficient of $x^n$ in the expansion of $x^2+bx+c^n$.Those $T n=T n1,1\ n=0,1,2,\ldots$ are the usual central trinomialcoefficients, and $T n3,2$ coincides with the Delannoy number$D n=\sum {k=0}^n\binom nk\binom{n+k}k$ in combinatorics. We investigatecongruences involving generalized central trinomial coefficientssystematically. Here are some typical results: For each $n=1,2,3,\ldots$ wehave $$\sum {k=0}^{n-1}2k+1T kb,c^2b^2-4c^{n-1-k}\equiv0\pmod{n^2}$$ andin particular $n^2\mid\sum {k=0}^{n-1}2k+1D k^2$; if $p$ is an odd prime then$$\sum {k=0}^{p-1}T k^2\equiv\left\frac{-1}p ight\ \pmod{p}\ \ \ { m and}\\ \ \sum {k=0}^{p-1}D k^2\equiv\left\frac 2p ight\ \pmod{p},$$ where $-$denotes the Legendre symbol. We also raise several conjectures some of whichinvolve parameters in the representations of primes by certain binary quadraticforms.



Author: Zhi-Wei Sun

Source: https://arxiv.org/







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