A377080 G.f.: Sum_{k>=1} x^(2*k^2) * Product_{j=1..k} (1 + x^j).

0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 3, 4, 4, 4, 5, 5, 5, 5, 4, 4

Offset: 0

Author

Vaclav Kotesovec, Oct 15 2024

Keywords

• nonn

Links

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

Crossrefs

Cf. A306734, A350890, A350893, A377081.

Cf. A230152.

Programs

Mathematica

nmax = 150; CoefficientList[Series[Sum[x^(2*k^2)*Product[1+x^j, {j, 1, k}], {k, 1, Sqrt[nmax/2]}], {x, 0, nmax}], x]

Formula

a(n) ~ (1+r) * exp(sqrt((40*log(r)^2 + 4*polylog(2, 1/(1+r)) - Pi^2/3)*n)) / (2*sqrt((4 + 5*r)*n)), where r = A230152 = 0.856674883854502874852324... is the real root of the equation r^4*(1+r) = 1.