Modular forms, hypergeometric functions and congruences

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_k(n)=\sum_{\substack{i_1,i_2,\ldots, i_k \ge0\\ i_1+i_2+\cdots+ i_k=n}}\binom{2i_1}{i_1}^2 \binom{2i_2}{i_2}^2\cdots \binom{2i_k}{i_k}^2, \quad \textrm{for } k,n\in\mathbb{N}. $$\end{document}