A376708 G.f.: Sum_{k>=0} x^(k*(k+1)) * Product_{j=1..k} 1/(1 - x^j)^3.

1, 0, 1, 3, 6, 10, 16, 24, 37, 55, 84, 124, 186, 270, 394, 561, 798, 1114, 1553, 2133, 2924, 3966, 5364, 7196, 9629, 12795, 16956, 22344, 29355, 38377, 50026, 64920, 84006, 108275, 139155, 178207, 227601, 289734, 367882, 465726, 588147

Offset: 0

Author

Vaclav Kotesovec, Oct 02 2024

Keywords

• nonn

Crossrefs

Cf. A003106, A064428, A376711.

Cf. A357471.

Programs

Mathematica

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

Formula

a(n) ~ r^(1/3) * (log(r)^2 + 3*polylog(2, 1-r))^(3/4) * exp(2*sqrt((log(r)^2 + 3*polylog(2, 1-r))*n)) / (4 * Pi^(3/2) * sqrt(2+r) * n^(5/4)), where r = 1 - A357471 = 0.430159709001946734... is the real root of the equation r^2 = (1-r)^3.