A376625 G.f.: Sum_{k>=0} x^(k*(k+1)/2) / Product_{j=1..k} (1 - x^(2*j))^2.

1, 1, 0, 3, 0, 5, 1, 9, 2, 13, 6, 20, 12, 27, 23, 39, 40, 51, 69, 70, 108, 92, 169, 125, 252, 166, 370, 227, 527, 307, 743, 425, 1021, 586, 1393, 816, 1867, 1132, 2481, 1577, 3256, 2184, 4247, 3019, 5479, 4149, 7036, 5670, 8966, 7698, 11377, 10386, 14356, 13915, 18060

Offset: 0

Author

Vaclav Kotesovec, Sep 30 2024

Keywords

• nonn

Links

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

Crossrefs

Cf. A000700, A072223, A179049, A340647, A376542, A376658.

Programs

Mathematica

nmax=80; CoefficientList[Series[Sum[x^(k*(k+1)/2)/Product[1-x^(2*j), {j, 1, k}]^2, {k, 0, Sqrt[2*nmax]}], {x, 0, nmax}], x]

Formula

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

a(n) ~ (r^(1/4) * sqrt(log(r)^2 + 2*polylog(2, sqrt(r))) / (2*Pi*sqrt(1 + 3*r^2))) * A376658^sqrt(n) / n, where r = A072223 = 0.52488859865640479389948613854128391569... is the smallest real root of the equation (1 - r^2)^2 = r.