Let N(1, 1)=1 and N(1,k)=0 if k<>1. Let N(s+1,k) = N(s,k) + N(s,k-1) + N(s,k+1).
Acting in such a way, we obtain the triangle
where only the nonzero terms are shown.
What is the explicit formula for N(s,k)? How to find it with Maple?