%I #21 Oct 10 2017 10:31:19
%S 1,5,23,77,244,677,1794,4411,10454,23597,51699,109378,225804,453893,
%T 893872,1723286,3265023,6078557,11148496,20146561,35935772,63287458,
%U 110186562,189715530,323335946,545666040,912512366,1512613763,2486819428,4056167621,6566647376
%N Expansion of Product_{k>0} ((1 - q^(3*k))^3*(1 - q^(6*k))^3)/((1 - q^k)^5*(1 - q^(2*k))^3).
%H Seiichi Manyama, <a href="/A293426/b293426.txt">Table of n, a(n) for n = 0..10000</a>
%H W. Y. C. Chen, K. Q. Ji, H.-T. Jin and E. Y. Y. Shen, <a href="https://doi.org/10.1016/j.jnt.2013.02.010">On the number of partitions with designated summands</a>, J. Number Theory, 133 (2013), 2929-2938.
%F a(n) = (1/3) * A077285(3*n+2).
%F 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
%t 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 *)
%t 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 *)
%o (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
%Y Cf. A077285 (PD(n)).
%K nonn
%O 0,2
%A _Seiichi Manyama_, Oct 09 2017
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