A377076 G.f.: Sum_{k>=0} x^(6*k^2) / Product_{j=1..6*k-1} (1 - x^j).

0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 10, 13, 18, 23, 30, 37, 47, 57, 70, 84, 101, 119, 142, 165, 194, 224, 260, 298, 344, 392, 449, 510, 582, 659, 750, 847, 962, 1087, 1233, 1393, 1581, 1787, 2029, 2297, 2610, 2958, 3365, 3819, 4348, 4942, 5630, 6404, 7302, 8310, 9475, 10787

Offset: 0

Author

Vaclav Kotesovec, Oct 15 2024

Keywords

• nonn

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Crossrefs

Column 6 of A350889.

Programs

Mathematica

nmax = 100; CoefficientList[Series[Sum[x^(6*k^2)/Product[1-x^j, {j, 1, 6*k-1}], {k, 1, Sqrt[nmax/6]}], {x, 0, nmax}], x]

Formula

a(n) ~ r^2 * (6*log(r)^2 + polylog(2, r^2))^(1/4) * exp(2*sqrt((6*log(r)^2 + polylog(2, r^2))*n)) / (2*sqrt(12*Pi*(3 - 2*r^2)) * n^(3/4)), where r = sqrt(((9 + sqrt(93))/2)^(1/3)/3^(2/3) - (2/(3*(9 + sqrt(93))))^(1/3)) = 0.82603135765418... is the positive real root of the equation r^2 = 1 - r^6.