Krawtchouk matrices

In mathematics, Krawtchouk matrices are matrices whose entries are values of Krawtchouk polynomials at nonnegative integer points.
The Krawtchouk matrix "K(N)" is an "(N+1)×(N+1)" matrix. Here are the first few examples: 
formula_1
<br>
formula_2
In general, for positive integer formula_3, the entries formula_4 are given via the generating function
where the row and column indices formula_6 and formula_7 run from formula_8 to formula_3.
These Krawtchouk polynomials are orthogonal with respect to symmetric binomial distributions, formula_10.