A377076 - OEIS

%I #8 Oct 15 2024 05:11:35

%S 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,

%T 165,194,224,260,298,344,392,449,510,582,659,750,847,962,1087,1233,

%U 1393,1581,1787,2029,2297,2610,2958,3365,3819,4348,4942,5630,6404,7302,8310,9475,10787

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

%H Vaclav Kotesovec, <a href="/A377076/b377076.txt">Table of n, a(n) for n = 0..10000</a>

%F 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.

%t 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]

%Y Column 6 of A350889.

%K nonn

%O 0,9

%A _Vaclav Kotesovec_, Oct 15 2024