A369520 Expansion of Product_{k>=1} 1/((1 - x^(k^2))*(1 - x^k)).

1, 2, 4, 7, 13, 21, 34, 52, 80, 119, 175, 251, 359, 504, 702, 965, 1320, 1785, 2401, 3200, 4245, 5589, 7324, 9535, 12364, 15944, 20478, 26175, 33338, 42279, 53438, 67283, 84454, 105642, 131764, 163826, 203149, 251185, 309799, 381079, 467666, 572520, 699342, 852314

Offset: 0

Author

Vaclav Kotesovec, Jan 25 2024

Keywords

• nonn

Comments

Convolution of A001156 and A000041.

a(n) is the number of pairs (Q(k), P(n-k)), 0<=k<=n, where Q(k) is a partition of k into squares and P(n-k) is a partition of n-k.

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
• Vaclav Kotesovec, Graph - the asymptotic ratio (250000 terms)

Crossrefs

Cf. A000041, A001156, A087153, A264393, A280204, A369519, A369579.

Programs

Mathematica

nmax=50; CoefficientList[Series[Product[1/(1-x^(k^2))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

a(n) ~ exp(Pi*sqrt(2*n/3) + 3^(1/4)*zeta(3/2)*n^(1/4)/2^(3/4) - 3*zeta(3/2)^2/(32*Pi)) / (2^(13/4) * 3^(3/4) * sqrt(Pi) * n^(5/4)).