A027999 Expansion of Product(1+q^m)^(m(m-1)/2); m=1..inf.
1, 0, 1, 3, 6, 13, 24, 49, 91, 181, 334, 632, 1163, 2138, 3880, 7006, 12531, 22279, 39369, 69078, 120597, 209282, 361405, 620829, 1061687, 1807014, 3062642, 5168784, 8688820, 14549659, 24274226, 40353748, 66854518, 110391391, 181695436, 298129605, 487706902
Offset: 0
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- Vaclav Kotesovec, Graph - the asymptotic ratio
Programs
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Maple
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(binomial(binomial(i, 2), j)*b(n-i*j, i-1), j=0..n/i))) end: a:= n-> b(n$2): seq(a(n), n=0..50); # Alois P. Heinz, Aug 03 2013 -
Mathematica
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[Binomial[Binomial[i, 2], j]*b[n-i*j, i-1], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Oct 13 2014, after Alois P. Heinz *)
Formula
a(n) ~ 7^(1/8) / (2^(47/24) * 15^(1/8) * n^(5/8)) * exp(-2025 * Zeta(3)^3 / (98*Pi^8) - 135*(15/7)^(1/4) * Zeta(3)^2 / (28*Pi^5) * n^(1/4) - 3*sqrt(15/7) * Zeta(3) / (2*Pi^2) * sqrt(n) + 2*(7/15)^(1/4) * Pi/3 * n^(3/4)), where Zeta(3) = A002117. - Vaclav Kotesovec, May 27 2015