OFFSET
0,2
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..1000
S. Benvenuti, B. Feng, A. Hanany and Y. H. He, Counting BPS operators in gauge theories: Quivers, syzygies and plethystics, arXiv:hep-th/0608050, 2006.
Vaclav Kotesovec, Graph - The asymptotic ratio
FORMULA
G.f.: exp( Sum_{n>0} (3*x^n - x^(2*n)) / (n*(1-x^n)^2) ).
a(n) ~ Zeta(3)^(7/18) * exp(1/6 - Pi^4/(864*Zeta(3)) + Pi^2 * n^(1/3)/(3 * 2^(5/3) * Zeta(3)^(1/3)) + 3 * (Zeta(3)/2)^(1/3) * n^(2/3)) / (A^2 * 2^(2/9) * 3^(1/2) * Pi * n^(8/9)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... . - Vaclav Kotesovec, Mar 07 2015
From Peter Bala, Jan 16 2025: (Start)
G.f.: 1/Product_{k >= 1} (1 - x^k)^(2*k+1).
G.f.: exp(Sum_{k >= 1} (2*sigma_2(k) + sigma_1(k))*x^k/k) = 1 + 3*x + 11*x^2 + 32*x^3 + 90*x^4 + 231*x^5 + .... (End)
MAPLE
with(numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d, j; if n=0 then 1 else add(add(d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:=etr(n-> 2*n+1): seq(a(n), n=0..50); # Vaclav Kotesovec, Mar 06 2015 after Alois P. Heinz
# alternative program
with(numtheory):
series(exp(add((2*sigma[2](k) + sigma[1](k))*x^k/k, k = 1..30)), x, 31):
seq(coeftayl(%, x = 0, n), n = 0..30); # Peter Bala, Jan 16 2025
MATHEMATICA
nmax=50; CoefficientList[Series[Product[1/(1-x^k)^(2*k+1), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Feb 27 2015 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Amihay Hanany (hanany(AT)mit.edu), Aug 25 2006
STATUS
approved