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A029863 Expansion of Product_{k >= 1} 1/(1-x^k)^c(k), where c(1), c(2), ... = 2 3 2 3 2 3 2 3 ....

%I A029863 #34 Dec 29 2024 09:06:28
%S A029863 1,2,6,12,27,50,98,172,310,522,888,1444,2357,3724,5882,9072,13957,
%T A029863 21082,31732,47072,69545,101540,147620,212516,304631,433054,613030,
%U A029863 861616,1206089,1677766,2324844,3203748,4398602,6009390,8181250
%N A029863 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 A029863 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 A029863 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 A029863 Songxin Liang & Jingzhong Zhang, <a href="https://doi.org/10.1007/BF02917106">A complete discrimination system for polynomials with complex coefficients and its automatic generation</a>, 1999, p. 2. the sequence Cn is the numbers of cases for root classification of an n-degree polynomial with complex coefficients.
%H A029863 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%F A029863 Euler transform of period 2 sequence [2, 3, ...].
%F A029863 a(n) ~ 5 * exp(sqrt(5*n/3)*Pi) / (48 * n^(3/2)). - _Vaclav Kotesovec_, Sep 20 2015
%F A029863 G.f.: Product_{k >= 1} 1/(1-x^k)^A010693(k-1). - _Georg Fischer_, Dec 10 2020
%e A029863 G.f. = 1 + 2*x + 6*x^2 + 12*x^3 + 27*x^4 + 50*x^5 + 98*x^6 + 172*x^7 + ...
%t A029863 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 A029863 (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 A029863 Cf. A002513, A010693, A085140, A262380.
%K A029863 nonn
%O A029863 0,2
%A A029863 _N. J. A. Sloane_