A120844 Number of multi-trace BPS operators for the quiver gauge theory of the orbifold C^2/Z_2.

1, 3, 11, 32, 90, 231, 576, 1363, 3141, 7003, 15261, 32468, 67788, 138892, 280103, 556302, 1089991, 2108332, 4030649, 7620671, 14261450, 26431346, 48544170, 88393064, 159654022, 286149924, 509137464, 899603036, 1579014769

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.

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

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

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

Cf. A005380, A253289, A255271, A255802, A255834.

Sequence in context: A089620 A144335 A202091 * A110958 A141211 A353065

Adjacent sequences: A120841 A120842 A120843 * A120845 A120846 A120847

Keyword

nonn

Author

Amihay Hanany (hanany(AT)mit.edu), Aug 25 2006

Status

approved