A376707 G.f.: Sum_{k>=0} x^(k*(k+1)/2) * Product_{j=1..k} 1/(1 - x^j)^3.

1, 1, 3, 7, 13, 24, 41, 70, 114, 186, 293, 459, 703, 1067, 1593, 2359, 3447, 4998, 7175, 10222, 14445, 20277, 28263, 39156, 53922, 73843, 100587, 136331, 183890, 246909, 330094, 439453, 582738, 769782, 1013169, 1328805, 1736942, 2263018, 2939280, 3806072, 4914221

Offset: 0

Author

Vaclav Kotesovec, Oct 02 2024

Keywords

• nonn

Crossrefs

Cf. A000009, A143184, A376710.

Cf. A263719.

Programs

Mathematica

nmax = 40; CoefficientList[Series[Sum[x^(k*(k+1)/2)/Product[1-x^j, {j, 1, k}]^3, {k, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x]

Formula

a(n) ~ r^(1/6) * (log(r)^2 + 6*polylog(2, 1-r))^(3/4) * exp(sqrt(2*(log(r)^2 + 6*polylog(2, 1-r))*n)) / (2^(11/4) * Pi^(3/2) * sqrt(1 + 2*r) * n^(5/4)), where r = 1 - A263719 = 0.3176721961719... is the real root of the equation r = (1-r)^3.