A193659 Q-residue of the triangle A094727, where Q=Pascal's triangle. (See Comments.)

1, 2, 8, 43, 265, 1832, 14160, 121771, 1157557, 12080436, 137505288, 1696841395, 22578385961, 322377704664, 4917809053032, 79840791037379, 1374705370985669, 25024307510421060, 480230285880218992

Offset: 0

Author

Clark Kimberling, Aug 02 2011

Keywords

• nonn

Comments

For the definition of Q-residue, see A193649.

Crossrefs

Cf. A193668, A094727, A193649.

Programs

Mathematica

q[n_, k_] := n + k + 1;(* A094727 *)
r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}]
p[n_, k_] := n!/(k! (n - k)!); (* Pascal's triangle *)
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 18}]    (* A193659 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}]  (* A193668 *)
TableForm[Table[p[n, k], {n, 0, 4}, {k, 0, n}]]

Formula

Conjecture: a(n) +(-n-5)*a(n-1) +2*(2*n+1)*a(n-2) +(-5*n+8)*a(n-3) +2*(n-3)

*a(n-4)=0. - R. J. Mathar, Feb 19 2015