A triangle sequence from a sum: t0(n,m)=(2 + PartitionsQ[n] - PartitionsQ[m] - PartitionsQ[n - m]); t1(n,k)=Sum[(-1)^j *t0[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}]; t(n,m)=If[n == 0, 1, t1(n, k) + t1(n, n - k)].

A156130

A triangle sequence from a sum: t0(n,m)=(2 + PartitionsQ[n] - PartitionsQ[m] - PartitionsQ[n - m]); t1(n,k)=Sum[(-1)^j *t0[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}]; t(n,m)=If[n == 0, 1, t1(n, k) + t1(n, n - k)].