cp's OEIS Frontend

A193654 Q-residue of the triangle p(n,k)=floor((n+1)/(n+k+2)/2), 0<=k<=n, where Q is the triangular array (t(i,j)) given by t(i,j)=1. (See Comments.)

%I A193654 #10 Feb 19 2015 14:48:30
%S A193654 1,7,28,94,275,765,2002,5116,12625,30715,73040,172026,398671,917497,
%T A193654 2086222,4718584,10573133,23592951,52254028
%N A193654 Q-residue of the triangle p(n,k)=floor((n+1)/(n+k+2)/2), 0<=k<=n, where Q is the triangular array (t(i,j)) given by t(i,j)=1.  (See Comments.)
%C A193654 For the definition of Q-residue, see A193649.
%F A193654 Conjecture: G.f.: ( -1-2*x+4*x^2+4*x^3-8*x^5 ) / ( (1+x)*(2*x+1)*(x-1)^2*(2*x-1)^3 ). - _R. J. Mathar_, Feb 19 2015
%t A193654 q[n_, k_] := 1;
%t A193654 r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}]
%t A193654 p[n_, k_] := Floor[(n + 1) (n + k + 2)/2]
%t A193654 v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
%t A193654 Table[v[n], {n, 0, 16}]    (* A193654 *)
%t A193654 TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
%t A193654 Table[r[k], {k, 0, 8}]  (* 2^k *)
%t A193654 TableForm[Table[p[n, k], {n, 0, 4}, {k, 0, n}]]
%Y A193654 Cf. A193649, A193655.
%K A193654 nonn
%O A193654 0,2
%A A193654 _Clark Kimberling_, Aug 02 2011