A291649 Expansion of Product_{k>=1} (1 + x^(k^2))^(k^2).

1, 1, 0, 0, 4, 4, 0, 0, 6, 15, 9, 0, 4, 40, 36, 0, 17, 71, 90, 36, 64, 100, 180, 144, 96, 274, 394, 300, 148, 740, 820, 480, 472, 1150, 1851, 1341, 1146, 1318, 3880, 3540, 1704, 3017, 6455, 7134, 3780, 7822, 9574, 12180, 10304, 12057, 19750, 22485, 20558, 15910, 43076, 43236, 31104, 33742, 66895

Offset: 0

Author

Ilya Gutkovskiy, Aug 28 2017

Keywords

• nonn

Comments

Number of partitions of n into distinct squares, where k^2 different parts of size k^2 are available (1a, 4a, 4b, 4c, 4d, ...).

Examples

			a(8) = 6 because we have [4a, 4b], [4a, 4c], [4a, 4d], [4b, 4c], [4b, 4d] and [4c, 4d].
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
• Index entries for related partition-counting sequences

Crossrefs

Cf. A000290, A026007, A027998, A033461, A262736, A281790, A284896, A291655, A291692.

Programs

Mathematica

nmax = 100; CoefficientList[Series[Product[(1 + x^k^2)^k^2, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; s = 1 + x; Do[s *= Sum[Binomial[k^2, j]*x^(j*k^2), {j, 0, Floor[nmax/k^2] + 1}]; s = Select[Expand[s], Exponent[#, x] <= nmax &];, {k, 2, nmax}]; CoefficientList[s, x] (* Vaclav Kotesovec, Aug 28 2017 *)

Formula

G.f.: Product_{k>=1} (1 + x^A000290(k))^A000290(k).

a(n) ~ exp(5 * 2^(-9/5) * 3^(-3/5) * (9-4*sqrt(2))^(1/5) * Pi^(1/5) * Zeta(5/2)^(2/5) * n^(3/5)) * 3^(1/5) * (2*sqrt(2)-1)^(1/5) * Zeta(5/2)^(1/5) / (2^(9/10) * sqrt(5) * Pi^(2/5) * n^(7/10)). - Vaclav Kotesovec, Aug 29 2017