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

0,2

Comments

Number of partitions of n where there are 2 kinds of odd parts and 3 kinds of even parts. - Ilya Gutkovskiy, Jan 17 2018

Links

Table of n, a(n) for n=0..34.

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.

N. J. A. Sloane, Transforms

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

Example

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

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 *)

Prog

(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))};

Crossrefs

Cf. A002513, A010693, A085140, A262380.

Sequence in context: A052971 A364423 A289443 * A091919 A059078 A335712

Adjacent sequences: A029860 A029861 A029862 * A029864 A029865 A029866

Keyword

nonn

Author

N. J. A. Sloane.

Status

approved