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A257949
The q-series expansion of an expression related to fermionic systems.
1
0, 0, 0, 12, 0, 60, 36, 168, 384, 396, 1620, 1452, 5388, 6396, 14616, 25860, 40128, 87108, 115992, 259236, 358860, 710220, 1096392, 1885080, 3216768, 4991700, 8916024, 13349448, 23633064, 35731944, 60638400, 94572072, 152913120, 245107764, 382072212, 620410980
OFFSET
0,4
COMMENTS
Conjecture: a linear combination of modular forms, possibly of different weights.
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?
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
David Kutasov, Finn Larsen, Partition Sums and Entropy Bounds in Weakly Coupled CFT, arXiv:hep-th/0009244, 2000.
David Kutasov, Finn Larsen, Partition Sums and Entropy Bounds in Weakly Coupled CFT, Journal of High Energy Physics, Volume 2001, JHEP01(2001)
MATHEMATICA
seriesQ[q_] = Exp[Sum[(-1)^(k + 1)/k (4 (q^k)^3)/(1 - (q^2)^k)^3, {k, 1, 200}]];
CoefficientList[Series[(960) q D[seriesQ[q], q], {q, 0, 100}]/960, q]
PROG
(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
CROSSREFS
Cf. A006352 (E_2), A004009 (E_4).
Sequence in context: A278711 A331911 A307841 * A375664 A375680 A376443
KEYWORD
nonn
AUTHOR
David McGady, May 14 2015
STATUS
approved