OFFSET
0,2
COMMENTS
Equals [1,2,3,...] * [1,0,4,0,10,0,20,...] * [1,0,0,6,0,0,21,...] * [1,0,0,0,8,0,0,0,36,...] * ... - Gary W. Adamson, Jul 06 2009
Number of pairs of planar partitions of u and v where u + v = n. - Joerg Arndt, Apr 22 2014
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vaclav Kotesovec)
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 19.
Paul Martin, Eric C. Rowell, and Fiona Torzewska, Classification of charge-conserving loop braid representations, arXiv:2301.13831 [math.QA], 2023.
FORMULA
G.f.: 1 / prod(k>=1, (1-x^k)^k )^2. - Joerg Arndt, Apr 22 2014
a(n) ~ Zeta(3)^(2/9) * exp(1/6 + 3*n^(2/3)*(Zeta(3)/2)^(1/3)) / (A^2 * 2^(1/18) * sqrt(3*Pi) * n^(13/18)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... . - Vaclav Kotesovec, Feb 27 2015
G.f.: exp(2*Sum_{k>=1} x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, May 29 2018
MAPLE
a:= proc(n) option remember; `if`(n=0, 1, 2*add(
a(n-j)*numtheory[sigma][2](j), j=1..n)/n)
end:
seq(a(n), n=0..30); # Alois P. Heinz, Mar 12 2015
MATHEMATICA
nn = 36; CoefficientList[Series[Product[1/(1 - x^i)^(2 i), {i, 1, nn}] , {x, 0, nn}], x] (* Geoffrey Critzer, Nov 29 2014 *)
PROG
(PARI) N=66; x='x+O('x^N); Vec(1/prod(k=1, N, (1-x^k)^k)^2) \\ Joerg Arndt, Apr 22 2014
CROSSREFS
KEYWORD
nonn
AUTHOR
Gary W. Adamson, Jun 20 2009
EXTENSIONS
Added more terms, Joerg Arndt, Apr 22 2014
STATUS
approved