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A parametric Catalan-type congruence for generalized central trinomial coefficients
Abstract
Let p>3 be a prime, and let b,c be integers with pnmid b+2c. We obtain an explicit congruence modulo p2 for the sum of binom2kkT2k(b,c2)/(4k(k+1)(b+2c)2k) from k=0 to (p−1)/2, where Tn(b,c) denotes the generalized central trinomial coefficient. This gives a parametric formula for the sum considered in a question of Wang and Cui. The proof uses a constant term representation and a logarithmic generating function to reduce the sum to two coefficients of an algebraic generating function. A differential identity and a coefficientwise polynomial congruence then yield the desired evaluation. The formula also applies when p divides b or b−2c.
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BibTeX
@misc{liu2026parametric,
title = {{A parametric Catalan-type congruence for generalized central trinomial coefficients}},
author = {Yuqi Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.31423},
eprint = {2609.31423},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.31423},
}
RIS
TY - UNPB
TI - A parametric Catalan-type congruence for generalized central trinomial coefficients
AU - Yuqi Liu
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.31423
UR - https://arxiv.org/abs/2609.31423
AB - Let p>3 be a prime, and let b,c be integers with pnmid b+2c. We obtain an explicit congruence modulo p^2 for the sum of binom2kkT_2k(b,c^2)/(4^k(k+1)(b+2c)^2k) from k=0 to (p−1)/2, where T_n(b,c) denotes the generalized central trinomial coefficient. This gives a parametric formula for the sum considered in a question of Wang and Cui. The proof uses a constant term representation and a logarithmic generating function to reduce the sum to two coefficients of an algebraic generating function. A differential identity and a coefficientwise polynomial congruence then yield the desired evaluation. The formula also applies when p divides b or b−2c.
ER -