Question 12
(POL) Let $n\ge k\ge0$ be integers. The numbers $c(n,k)$ are defined as follows:
$$c(n,0)=c(n,n)=1\qquad(n\ge0);$$
$$c(n+1,k)=2^kc(n,k)+c(n,k-1)\qquad(n\ge k\ge1).$$
Prove that $c(n,k)=c(n,n-k)$ for all $n\ge k\ge0$.
$$c(n,0)=c(n,n)=1\qquad(n\ge0);$$
$$c(n+1,k)=2^kc(n,k)+c(n,k-1)\qquad(n\ge k\ge1).$$
Prove that $c(n,k)=c(n,n-k)$ for all $n\ge k\ge0$.