A161870 Convolution square of A000219.

1, 2, 7, 18, 47, 110, 258, 568, 1237, 2600, 5380, 10870, 21652, 42350, 81778, 155676, 292964, 544846, 1003078, 1828128, 3301952, 5911740, 10499385, 18502582, 32371011, 56240816, 97073055, 166497412, 283870383, 481212656, 811287037, 1360575284, 2270274785, 3769835178, 6230705170, 10251665550, 16794445441

Offset: 0

Author

Gary W. Adamson, Jun 20 2009

Keywords

• nonn

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.

Crossrefs

Cf. A000219.

Column k=2 of A255961.

Programs

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 *)

PARI

N=66;x='x+O('x^N); Vec(1/prod(k=1,N,(1-x^k)^k)^2) \\ Joerg Arndt, Apr 22 2014

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

Extensions

Added more terms, Joerg Arndt, Apr 22 2014