A120844 - OEIS

%I #28 Jan 16 2025 19:37:21

%S 1,3,11,32,90,231,576,1363,3141,7003,15261,32468,67788,138892,280103,

%T 556302,1089991,2108332,4030649,7620671,14261450,26431346,48544170,

%U 88393064,159654022,286149924,509137464,899603036,1579014769

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

%H Vaclav Kotesovec, <a href="/A120844/b120844.txt">Table of n, a(n) for n = 0..1000</a>

%H S. Benvenuti, B. Feng, A. Hanany and Y. H. He, <a href="http://arXiv.org/abs/hep-th/0608050">Counting BPS operators in gauge theories: Quivers, syzygies and plethystics</a>, arXiv:hep-th/0608050, 2006.

%H Vaclav Kotesovec, <a href="/A120844/a120844.jpg">Graph - The asymptotic ratio</a>

%F G.f.: exp( Sum_{n>0} (3*x^n - x^(2*n)) / (n*(1-x^n)^2) ).

%F 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

%F From _Peter Bala_, Jan 16 2025: (Start)

%F G.f.: 1/Product_{k >= 1} (1 - x^k)^(2*k+1).

%F 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)

%p 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_

%p # alternative program

%p with(numtheory):

%p series(exp(add((2*sigma[2](k) + sigma[1](k))*x^k/k, k = 1..30)), x, 31):

%p seq(coeftayl(%, x = 0, n), n = 0..30); # _Peter Bala_, Jan 16 2025

%t nmax=50; CoefficientList[Series[Product[1/(1-x^k)^(2*k+1),{k,1,nmax}],{x,0,nmax}],x] (* _Vaclav Kotesovec_, Feb 27 2015 *)

%Y Cf. A000203, A001157, A005380, A217093, A253289, A255271, A255802, A255803, A255834, A255835, A255836, A363601, A363602.

%K nonn,easy

%O 0,2

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