A385520 Expansion of Product_{k>0} ((1 - q^(2*k))*(1 - q^(6*k))^3)/((1 - q^k)*(1 - q^(3*k))*(1 - q^(4*k))*(1 - q^(12*k))).

1, 1, 1, 3, 4, 5, 6, 9, 13, 16, 20, 27, 36, 44, 54, 69, 88, 107, 130, 162, 200, 240, 288, 351, 426, 507, 602, 723, 864, 1019, 1200, 1422, 1681, 1968, 2300, 2700, 3160, 3674, 4266, 4965, 5768, 6665, 7692, 8892, 10260, 11792, 13536, 15552, 17844, 20407

Offset: 0

Author

James Sellers, Jul 01 2025

Keywords

• nonn

Comments

a(n) is the number of partitions of the integer n wherein each even part can appear at most twice, while each odd part can appear once, thrice, or four times.

Also, a(n) is the unsigned version of the sequence given in A293306 (this can be seen by replacing q by -q in the generating function).

Examples

			For n = 4, the a(4) = 4 partitions are 4, 3+1, 2+2, and 1+1+1+1.  Note that there is one other partition of 4 which is NOT counted by a(4); that is the partition 2+1+1. This partition is NOT counted by a(4) because the odd part 1 appears twice, and this is not allowed from the description given above.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• Shane Chern, Dennis Eichhorn, Shishuo Fu, and James A. Sellers, Convolutive sequences, I: Through the lens of integer partition functions, arXiv:2507.10965 [math.CO], 2025. See pp. 4, 15.

Crossrefs

Cf. A293306.

Programs

Maple

p:=product((1-q^(2*k))*(1-q^(6*k))^3/((1-q^k)*(1-q^(3*k))*(1-q^(4*k))*(1-q^(12*k))), k=1..1000): s:=series(p,q,1000):

Mathematica

nmax = 50; CoefficientList[Series[Product[(1 - x^(2*k)) * (1 - x^(6*k))^3 / ((1 - x^k) * (1 - x^(3*k)) * (1 - x^(4*k)) * (1 - x^(12*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 02 2025 *)

PARI

my(N=50,q='q+O('q^N)); Vec((eta(-q)*eta(-q^3))/eta(q^2)^2) \\ Joerg Arndt, Jul 02 2025

Formula

G.f.: Product_{k>0} ((1 - q^(2*k))*(1 - q^(6*k))^3)/((1 - q^k)*(1 - q^(3*k))*(1 - q^(4*k))*(1 - q^(12*k))).

G.f.: (eta(-q)*eta(-q^3))/eta(q^2)^2.

a(n) ~ exp(2*Pi*sqrt(n)/3) / (6*n^(3/4)). - Vaclav Kotesovec, Jul 02 2025