A293426 Expansion of Product_{k>0} ((1 - q^(3*k))^3*(1 - q^(6*k))^3)/((1 - q^k)^5*(1 - q^(2*k))^3).

1, 5, 23, 77, 244, 677, 1794, 4411, 10454, 23597, 51699, 109378, 225804, 453893, 893872, 1723286, 3265023, 6078557, 11148496, 20146561, 35935772, 63287458, 110186562, 189715530, 323335946, 545666040, 912512366, 1512613763, 2486819428, 4056167621, 6566647376

Offset

0,2

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000

W. Y. C. Chen, K. Q. Ji, H.-T. Jin and E. Y. Y. Shen, On the number of partitions with designated summands, J. Number Theory, 133 (2013), 2929-2938.

Formula

a(n) = (1/3) * A077285(3*n+2).

a(n) ~ 5^(3/4) * exp(sqrt(10*n/3)*Pi) / (2^(11/4) * 3^(15/4) * n^(5/4)). - Vaclav Kotesovec, Oct 09 2017

Mathematica

nmax = 40; CoefficientList[Series[Product[((1 - x^(3*k))^3 * (1 - x^(6*k))^3) / ((1 - x^k)^5 * (1 - x^(2*k))^3), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 09 2017 *)
max = 30; QP = QPochhammer; s = QP[q^6]/(QP[q]*QP[q^2]*QP[q^3]) + O[q]^(3 max + 3); (1/3)*Table[CoefficientList[s, q][[3*n + 3]], {n, 0, max}] (* Jean-François Alcover, Oct 10 2017, from 1st formula *)

Prog

(PARI) m = 40; Vec(prod(k=1, m, ((1 - q^(3*k))^3*(1 - q^(6*k))^3)/((1 - q^k)^5*(1 - q^(2*k))^3)) + O(q^m)) \\ Michel Marcus, Oct 10 2017

Crossrefs

Cf. A077285 (PD(n)).

Sequence in context: A121868 A111584 A225266 * A267202 A211809 A211922

Adjacent sequences: A293423 A293424 A293425 * A293427 A293428 A293429

Keyword

nonn

Author

Seiichi Manyama, Oct 09 2017

Status

approved