A193658 Q-residue of the triangle A051162, where Q is the triangular array (t(i,j)) given by t(i,j)=1. (See Comments.)

1, 1, 3, 15, 83, 503, 3403, 25807, 218451, 2049687, 21160667, 238690847, 2923054435, 38641535143, 548635554795, 8328494925615, 134634766604915, 2309386642312631, 41897258229334267, 801610384425038911, 16132033041827096451

Offset: 0

Author

Clark Kimberling, Aug 02 2011

Keywords

• nonn

Comments

For the definition of Q-residue, see A193649.

Crossrefs

Cf. A051162, A193649.

Programs

Mathematica

q[n_, k_] := n + k; (* A051162 *)
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)!);
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 20}]    (* A193658 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 20}]  (* A001340 *)
TableForm[Table[p[n, k], {n, 0, 4}, {k, 0, n}]]

Formula

Conjecture: a(n) +(-n-4)*a(n-1) +(4*n-1)*a(n-2) +5*(-n+2)*a(n-3) +2*(n-3)*a(n-4)=0. - R. J. Mathar, Feb 19 2015