Enumeration of sequences by length and spread

AbstractLet a be a fixed nonnegative integer and let Nj(n, k) denote the number of sequences (a1, a2,…, an) such that 0≤as≤a(1≤s≤n),an=j ∑s=1n−1|as−as+1|=kThe numbers Nj(n, k) are evaluated by means of the generating functions Fj(x,y)=∑n=1∞∑k=0∞Nj(n,k)xnykMoreover it is proved that Nj(n, k) is a polynomial in n of degree k