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Expansion of Product_{k >= 1} 1/(1-x^k)^c(k), where c(1), c(2), ... = 2 3 2 3 2 3 2 3 ....
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%I #28 Dec 10 2020 17:20:40

%S 1,2,6,12,27,50,98,172,310,522,888,1444,2357,3724,5882,9072,13957,

%T 21082,31732,47072,69545,101540,147620,212516,304631,433054,613030,

%U 861616,1206089,1677766,2324844,3203748,4398602,6009390,8181250

%N Expansion of Product_{k >= 1} 1/(1-x^k)^c(k), where c(1), c(2), ... = 2 3 2 3 2 3 2 3 ....

%C Number of partitions of n where there are 2 kinds of odd parts and 3 kinds of even parts. - _Ilya Gutkovskiy_, Jan 17 2018

%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], 2015-2016, p. 16.

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>

%F Euler transform of period 2 sequence [2, 3, ...].

%F a(n) ~ 5 * exp(sqrt(5*n/3)*Pi) / (48 * n^(3/2)). - _Vaclav Kotesovec_, Sep 20 2015

%F G.f.: Product_{k >= 1} 1/(1-x^k)^A010693(k-1). - _Georg Fischer_, Dec 10 2020

%e G.f. = 1 + 2*x + 6*x^2 + 12*x^3 + 27*x^4 + 50*x^5 + 98*x^6 + 172*x^7 + ...

%t nmax = 50; CoefficientList[Series[Product[1/((1 + x^k)*(1 - x^k)^3), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 20 2015 *)

%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( 1 / (eta(x + A)^2 * eta(x^2 + A)), n))};

%Y Cf. A002513, A010693, A085140, A262380.

%K nonn

%O 0,2

%A _N. J. A. Sloane_.