A298311 Expansion of Product_{k>=1} 1/((1 - x^(2*k))*(1 - x^(2*k-1))^3).

1, 3, 7, 16, 32, 61, 112, 197, 336, 560, 912, 1456, 2287, 3536, 5392, 8123, 12096, 17824, 26016, 37632, 53984, 76848, 108601, 152432, 212592, 294704, 406201, 556864, 759488, 1030784, 1392496, 1872784, 2508048, 3345184, 4444384, 5882747, 7758736, 10197712, 13358944, 17444256, 22708719

Offset: 0

Author

Ilya Gutkovskiy, Jan 17 2018

Keywords

• nonn
• easy

Comments

Number of partitions of n where there are 3 kinds of odd parts.

Convolution of the sequences A000009 and A015128.

Links

• Michael D. Hirschhorn and James A. Sellers, A Family of Congruences Modulo 7 for Partitions with Monochromatic Even Parts and Multi-Colored Odd Parts, arXiv:2507.09752 [math.CO], 2025. See p. 2.
• Index entries for related partition-counting sequences

Crossrefs

Cf. A000009, A000041, A000716, A015128, A029862, A029863, A160461, A182818, A278690.

Programs

Mathematica

nmax = 40; CoefficientList[Series[Product[1/((1 - x^(2 k)) (1 - x^(2 k - 1))^3), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Product[(1 + x^k)^2/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} 1/((1 - x^(2*k))*(1 - x^(2*k-1))^3).

G.f.: Product_{k>=1} (1 + x^k)^2/(1 - x^k).

a(n) ~ exp(2*Pi*sqrt(n/3)) / (2^(5/2)*sqrt(3)*n). - Vaclav Kotesovec, Apr 08 2018

G.f.: 1/Product_{n > = 1} ( 1 - x^(n/gcd(n,k)) ) for k = 4. Cf. A000041 (k = 1), A015128 (k = 2), A278690 (k = 3) and A160461 (k = 5). - Peter Bala, Nov 17 2020