OFFSET
0,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..10000
A. Brini and A. Tanzini, Exact results for topological strings on resolved Y(p,q) singularities, arXiv:0804.2598 [hep-th], 2008-2009, p. 40, equation (6.82).
FORMULA
Expansion of (2 * E(k) - k'^2 * K(k)) * K(k) * (2/Pi)^2 in powers of q.
Expansion of (E(k) + k' * K(k)) * K(k) * (2/Pi)^2 / 2 in powers of q^4.
Expansion of (4 * P(q^2) - P(q)) / 3 in powers of q where P() is a Ramanujan Lambert series.
G.f.: 1 + 8 * Sum_{k>0} x^k / (1 + x^k)^2.
G.f.: 1 - 8 * Sum_{k>0} k * (-x)^k / (1 - x^k).
G.f.: 1 + 8 * Sum_{k>0} k * x^k * (1 - 3*x^k) / (1 - x^(2*k)).
Expansion of 8*q*theta_2(0,q)' / theta_2(0,q) in powers of q=exp(2*Pi*i*tau), where theta_2(z,q) is a Jacobi theta function. - Sander Mack-Crane, Nov 07 2013
EXAMPLE
G.f. = 1 + 8*q - 8*q^2 + 32*q^3 - 40*q^4 + 48*q^5 - 32*q^6 + 64*q^7 - 104*q^8 + ...
MATHEMATICA
a[n_] := SeriesCoefficient[8 q D[Series[EllipticTheta[2, 0, q^(1/2)], {q, 0, n + 1}], q] / Series[EllipticTheta[2, 0, q^(1/2)], {q, 0, n + 1}], {q, 0, n}] (* Sander Mack-Crane, Nov 07 2013 *)
a[ n_] := If[ n < 1, Boole[n == 0], -8 DivisorSum[ n, # (-1)^# &]]; (* Michael Somos, Jun 02 2015 *)
a[ n_] := SeriesCoefficient[ With[{f = EllipticTheta[ 2, 0, q^(1/2)]}, 8 q D[f + O[q]^(n + 1), q] / f], {q, 0, n}]; (* Michael Somos, Jun 02 2015 *)
CoefficientList[Series[(2/Pi) EllipticE[InverseEllipticNomeQ[Sqrt[q]]] EllipticTheta[3, 0, Sqrt[q]]^2, {q, 0, 40}], q] (* Jan Mangaldan, Jul 07 2020 *)
PROG
(PARI) {a(n) = if( n<1, n==0, -8 * sumdiv(n, d, (-1)^d * d))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Sep 15 2006
STATUS
approved