A318026 Expansion of Product_{k>=1} 1/((1 - x^k)*(1 - x^(3*k))).

1, 1, 2, 4, 6, 9, 16, 22, 33, 50, 70, 98, 143, 193, 266, 368, 493, 659, 892, 1170, 1543, 2035, 2642, 3422, 4448, 5694, 7294, 9334, 11839, 14982, 18968, 23812, 29868, 37410, 46598, 57924, 71953, 88913, 109728, 135212, 165991, 203407, 248986, 303706, 369939, 449967, 545820, 661038, 799629

Offset: 0

Author

Ilya Gutkovskiy, Aug 13 2018

Keywords

• nonn

Comments

Convolution of A000041 and A035377.

Convolution of A000712 and A137569.

Convolution inverse of A030203.

Number of partitions of n if there are 2 kinds of parts that are multiples of 3.

Examples

			a(4) = 6 because we have [4], [3, 1], [3', 1], [2, 2], [2, 1, 1] and [1, 1, 1, 1].
		

Links

• Zakir Ahmed, Nayandeep Deka Baruah, Manosij Ghosh Dastidar, New congruences modulo 5 for the number of 2-color partitions, Journal of Number Theory, Volume 157, December 2015, Pages 184-198.
• Index entries for sequences related to partitions

Crossrefs

Cf. A000041, A000712, A002513, A030203, A030206, A035377, A112175, A137569, A318027, A318028.

Programs

Maple

a:=series(mul(1/((1-x^k)*(1-x^(3*k))),k=1..55),x=0,49): seq(coeff(a,x,n),n=0..48); # Paolo P. Lava, Apr 02 2019

Mathematica

nmax = 48; CoefficientList[Series[Product[1/((1 - x^k) (1 - x^(3 k))), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[1/(QPochhammer[x] QPochhammer[x^3]), {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[Exp[Sum[x^k (1 + x^k + 2 x^(2 k))/(k (1 - x^(3 k))), {k, 1, nmax}]], {x, 0, nmax}], x]
Table[Sum[PartitionsP[k] PartitionsP[n - 3 k], {k, 0, n/3}], {n, 0, 48}]

Formula

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

a(n) ~ exp(2*sqrt(2*n)*Pi/3) / (3 * 2^(5/4) * n^(5/4)). - Vaclav Kotesovec, Aug 14 2018