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%I #18 Mar 15 2018 18:27:56
%S 0,0,0,12,0,60,36,168,384,396,1620,1452,5388,6396,14616,25860,40128,
%T 87108,115992,259236,358860,710220,1096392,1885080,3216768,4991700,
%U 8916024,13349448,23633064,35731944,60638400,94572072,152913120,245107764,382072212,620410980
%N The q-series expansion of an expression related to fermionic systems.
%C Conjecture: a linear combination of modular forms, possibly of different weights.
%C This sequence comes from physics and seems to be connected to modular forms. It is the q-series expansion of the derivative of the log of the middle expression Eq. (2.10) for fermions in the reference below. Similar expressions for bosonic systems in Eq. (2.10) yield (I) the Eisenstein series of weight 4 when the bosons are pure scalars, or (II) a difference of Eisenstein series of weight 4 and weight 2 when the bosons are fermions. I suspect that the fermionic series also has a name. What is the name?
%H Charles R Greathouse IV, <a href="/A257949/b257949.txt">Table of n, a(n) for n = 0..10000</a>
%H David Kutasov, Finn Larsen, <a href="http://arxiv.org/abs/hep-th/0009244">Partition Sums and Entropy Bounds in Weakly Coupled CFT</a>, arXiv:hep-th/0009244, 2000.
%H David Kutasov, Finn Larsen, <a href="http://dx.doi.org/10.1088/1126-6708/2001/01/001">Partition Sums and Entropy Bounds in Weakly Coupled CFT</a>, Journal of High Energy Physics, Volume 2001, JHEP01(2001)
%t seriesQ[q_] = Exp[Sum[(-1)^(k + 1)/k (4 (q^k)^3)/(1 - (q^2)^k)^3, {k, 1, 200}]];
%t CoefficientList[Series[(960) q D[seriesQ[q], q], {q, 0, 100}]/960, q]
%o (PARI) list(lim)=my(q='q+O('q^(lim\1-1))); concat([0,0,0], Vec(exp(sum(k=1,lim\2+1, -4*(-1)^k/k*q^(3*k)/(1-q^(2*k))^3))')) \\ _Charles R Greathouse IV_, May 14 2015
%Y Cf. A006352 (E_2), A004009 (E_4).
%K nonn
%O 0,4
%A _David McGady_, May 14 2015
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