Congruences for certain binomial sums

We exploit the properties of Legendre polynomials defined by the contour integral \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm{P}}_n}(z) = {(2{\rm{\pi i}})^{ - 1}}\oint {{{(1 - 2tz + {t^2})}^{ - 1/2}}{t^{ - n - 1}}{\rm{d}}t} $$\end{document}, where the contour encloses the origin and is traversed in the counterclockwise direction, to obtain congruences of certain sums of central binomial coefficients. More explicitly, by comparing various expressions of the values of Legendre polynomials, it can be proved that for any positive integer r, a prime p ⩾5 and n = rp2 − 1, we have \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum\limits_{k = 0}^{\left\lfloor {n/2} \right\rfloor } {(_k^{2k} ) \equiv 0,1} $$\end{document} or −1 (mod p2), depending on the value of r (mod 6).