A369570 Expansion of Product_{k>=1} (1 + x^(k^2)) * (1 + x^k).

1, 2, 2, 3, 5, 7, 9, 12, 15, 20, 27, 33, 41, 52, 65, 80, 99, 120, 145, 177, 213, 255, 305, 363, 430, 511, 604, 709, 833, 976, 1141, 1331, 1547, 1793, 2079, 2406, 2775, 3197, 3676, 4221, 4841, 5541, 6330, 7225, 8235, 9372, 10655, 12094, 13710, 15529, 17568, 19848

Offset: 0

Author

Vaclav Kotesovec, Jan 26 2024

Keywords

• nonn

Comments

Convolution of A033461 and A000009.

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

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
• Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)

Crossrefs

Cf. A000009, A033461, A280204, A280276.

Programs

Mathematica

nmax = 100; CoefficientList[Series[Product[(1+x^(k^2))*(1+x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

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