A104253 Row sums of triangle in A116925.

1, 3, 6, 11, 21, 45, 113, 339, 1221, 5273, 27237, 167985, 1235820, 10838397, 113281002, 1410702627, 20928310905, 369834091857, 7784253038081, 195135698311989, 5825657474768916, 207120610510791805, 8769156584345509398, 442116458092151729925, 26542966216935028587896

Offset: 0

Author

N. J. A. Sloane, Sep 08 2006

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..162

Crossrefs

Cf. A116925.

Programs

Mathematica

Table[Sum[1 + Sum[Product[Binomial[n-1, n - s + j]/Binomial[n-1, j], {j, 0, k-1}], {k, 1, s}], {s, 0, n}], {n, 0, 25}] (* Vaclav Kotesovec, Apr 02 2021 *)
Table[BarnesG[1 + n] * Sum[BarnesG[1 + k] * BarnesG[1 + n - s] * BarnesG[1 - k + s] / (BarnesG[1 - k + n] * BarnesG[1 + k + n - s] * BarnesG[1 + s]), {s, 0, n}, {k, 0, s}], {n, 0, 20}] (* Vaclav Kotesovec, Apr 02 2021 *)

Formula

a(n) ~ c * BarnesG(n/3 + 1)^3 * BarnesG(n+1) / BarnesG(2*n/3 + 1)^3 ~ c * exp(1/12) * 3^(n^2/2) / (A * n^(1/12) * 2^(2*n^2/3 - 1/4)), where c = 5.2335188744705752675068634418929940491557563366762252523140713171090086689943... and A is the Glaisher-Kinkelin constant A074962. - Vaclav Kotesovec, Apr 02 2021