A255610 G.f.: Product_{k>=1} 1/(1-x^k)^(3*k).
1, 3, 12, 37, 111, 303, 804, 2022, 4950, 11715, 27081, 61083, 135112, 293142, 625620, 1314267, 2722323, 5564172, 11234865, 22424904, 44284545, 86573147, 167648418, 321746907, 612274678, 1155782109, 2165116416, 4026391221, 7435806048, 13641093684, 24865920932
Offset: 0
Keywords
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.
- Vaclav Kotesovec, Graph - The asymptotic ratio (250000 terms)
- Eric Weisstein's World of Mathematics, Plane Partition
- Wikipedia, Plane partition
Programs
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Maple
a:= proc(n) option remember; `if`(n=0, 1, 3*add( a(n-j)*numtheory[sigma][2](j), j=1..n)/n) end: seq(a(n), n=0..30); # Alois P. Heinz, Mar 11 2015 -
Mathematica
nmax=50; CoefficientList[Series[Product[1/(1-x^k)^(3*k),{k,1,nmax}],{x,0,nmax}],x]
Formula
G.f.: Product_{k>=1} 1/(1-x^k)^(3*k).
a(n) ~ Zeta(3)^(1/4) * exp(1/4 + 2^(-2/3) * 3^(4/3) * Zeta(3)^(1/3) * n^(2/3)) / (A^3 * 6^(1/4) * sqrt(Pi) * n^(3/4)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... . - Vaclav Kotesovec, Feb 28 2015
More precise asymptotics: a(n) ~ Zeta(3)^(1/4) * exp(1/4 + 2^(-2/3) * 3^(4/3) * Zeta(3)^(1/3) * n^(2/3)) / (A^3 * 6^(1/4) * sqrt(Pi) * n^(3/4)) * (1 - c/n^(2/3)), where c = 0.21774822... . - Vaclav Kotesovec, Oct 15 2015
G.f.: exp(3*Sum_{k>=1} x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, May 29 2018
Extensions
New name from Vaclav Kotesovec, Mar 12 2015