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

%I A193658 #8 Feb 19 2015 14:49:29
%S A193658 1,1,3,15,83,503,3403,25807,218451,2049687,21160667,238690847,
%T A193658 2923054435,38641535143,548635554795,8328494925615,134634766604915,
%U A193658 2309386642312631,41897258229334267,801610384425038911,16132033041827096451
%N A193658 Q-residue of the triangle A051162, where Q is the triangular array (t(i,j)) given by t(i,j)=1.  (See Comments.)
%C A193658 For the definition of Q-residue, see A193649.
%F A193658 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
%t A193658 q[n_, k_] := n + k; (* A051162 *)
%t A193658 r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}]
%t A193658 p[n_, k_] := n!/(k! (n - k)!);
%t A193658 v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
%t A193658 Table[v[n], {n, 0, 20}]    (* A193658 *)
%t A193658 TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
%t A193658 Table[r[k], {k, 0, 20}]  (* A001340 *)
%t A193658 TableForm[Table[p[n, k], {n, 0, 4}, {k, 0, n}]]
%Y A193658 Cf. A051162, A193649.
%K A193658 nonn
%O A193658 0,3
%A A193658 _Clark Kimberling_, Aug 02 2011