A193655 -id:A193655 - cp's OEIS frontend

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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.)

Original entry on oeis.org

1, 7, 28, 94, 275, 765, 2002, 5116, 12625, 30715, 73040, 172026, 398671, 917497, 2086222, 4718584, 10573133, 23592951, 52254028

Offset: 0

Clark Kimberling, Aug 02 2011

nonn

For the definition of Q-residue, see A193649.

Cf. A193649, A193655.

q[n_, k_] := 1;
r[0] = 1; r[k_] := Sum[q[k - 1, i] r[k - 1 - i], {i, 0, k - 1}]
p[n_, k_] := Floor[(n + 1) (n + k + 2)/2]
v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}]
Table[v[n], {n, 0, 16}]    (* A193654 *)
TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]]
Table[r[k], {k, 0, 8}]  (* 2^k *)
TableForm[Table[p[n, k], {n, 0, 4}, {k, 0, n}]]

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

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