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Supercongruences involving Motzkin numbers and central trinomial coefficients
Ji-Cai Liu
Let and denote the th Motzkin number and the th central trinomial coefficient respectively. We prove that for any prime , \begin{align*} &\sum_{k=0}^{p-1}…
Further results on the divisibility of -trinomial coefficients
Ji-Cai Liu, Wei-Wei Qi
We study divisibility for the -trinomial coefficients , and , which were first introduced by Andrews and Baxter. In particular, we completel…
Proof of a conjecture of Z.-W. Sun on the divisibility of a triple sum
Victor J. W. Guo, Ji-Cai Liu
The numbers and are defined as \begin{align*} R_n=\sum_{k=0}^{n}{n+k\choose 2k}{2k\choose k}\frac{1}{2k-1},\ \text{and}\ W_n=\sum_{k=0}^{n}{n+k\choose 2k}{2k\choose k}\…
Proof of some conjectures of Z.-W. Sun on the divisibility of certain double-sums
Victor J. W. Guo, Ji-Cai Liu
Z.-W. Sun introduced three kinds of numbers: \begin{align*}S_n=\sum_{k=0}^{n}{n\choose k}^2{2k\choose k}(2k+1),\qquad s_n=\sum_{k=0}^{n}{n\choose k}^2{2k\choose k}\frac{1}{2k-1}, \…