New identities for binary Krawtchouk polynomials, binomial coefficients   and Catalan numbers
  We obtain new combinatorial identities for integral values of binary Krawtchouk polynomials $K^{2m}_p(x)$, $0\le p\le 2m$, by computing the characters of the $p$-exterior representations on certain elements of order 2 of $\mathrm{SO}(2m)$. From this identities, we deduce several new relations for binomial coefficients and Catalan numbers. 
Ricardo A. Podest\'a
math.CO