A193655 Q-residue of the triangle p(n,k)=floor(1/2+(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.)
1, 7, 29, 94, 280, 765, 2023, 5116, 12710, 30715, 73381, 172026, 400036, 917497, 2091683, 4718584, 10594978, 23592951, 52341409, 115343350, 253405856
Offset: 0
Keywords
Programs
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Mathematica
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[1/2 + (n + 1) (n + k + 2)/2] v[n_] := Sum[p[n, k] r[n - k], {k, 0, n}] Table[v[n], {n, 0, 20}] (* A193655 *) TableForm[Table[q[i, k], {i, 0, 4}, {k, 0, i}]] Table[r[k], {k, 0, 8}] TableForm[Table[p[n, k], {n, 0, 4}, {k, 0, n}]]
Formula
Conjecture: G.f.: ( -1-2*x+3*x^2+9*x^3-8*x^4-4*x^5 ) / ( (1+x)*(2*x+1)*(x-1)^2*(2*x-1)^3 ). - R. J. Mathar, Feb 19 2015
Comments