A257949 The q-series expansion of an expression related to fermionic systems.

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 * A077351 A370018 A119530

Adjacent sequences: A257946 A257947 A257948 * A257950 A257951 A257952

Keyword

nonn

Author

David McGady, May 14 2015

Status

approved