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 ....
1, 2, 6, 12, 27, 50, 98, 172, 310, 522, 888, 1444, 2357, 3724, 5882, 9072, 13957, 21082, 31732, 47072, 69545, 101540, 147620, 212516, 304631, 433054, 613030, 861616, 1206089, 1677766, 2324844, 3203748, 4398602, 6009390, 8181250
Offset: 0
Keywords
Examples
G.f. = 1 + 2*x + 6*x^2 + 12*x^3 + 27*x^4 + 50*x^5 + 98*x^6 + 172*x^7 + ...
Links
- Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016, p. 16.
- Songxin Liang & Jingzhong Zhang, A complete discrimination system for polynomials with complex coefficients and its automatic generation, 1999, p. 2. the sequence Cn is the numbers of cases for root classification of an n-degree polynomial with complex coefficients.
- N. J. A. Sloane, Transforms
Programs
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Mathematica
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 *) -
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))};
Formula
Euler transform of period 2 sequence [2, 3, ...].
a(n) ~ 5 * exp(sqrt(5*n/3)*Pi) / (48 * n^(3/2)). - Vaclav Kotesovec, Sep 20 2015
G.f.: Product_{k >= 1} 1/(1-x^k)^A010693(k-1). - Georg Fischer, Dec 10 2020
Comments