A122858 Expansion of E(k) * K(k) * (2/Pi)^2 in powers of q^2 where E(), K() are complete elliptic integrals and the nome q = exp( -Pi * K(k') / K(k)).

1, 8, -8, 32, -40, 48, -32, 64, -104, 104, -48, 96, -160, 112, -64, 192, -232, 144, -104, 160, -240, 256, -96, 192, -416, 248, -112, 320, -320, 240, -192, 256, -488, 384, -144, 384, -520, 304, -160, 448, -624, 336, -256, 352, -480, 624, -192, 384, -928, 456

Offset

0,2

Comments

Ramanujan Lambert series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).

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)).

a(n) = 8 * A002129(n) unless n=0. a(n) = (-1)^n * A143336(n).

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

Conjecture: -3 A122858(n) - A229616(n) + 4 A282031(n) = 0 for all n. - Thomas Baruchel, Jun 23 2018

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

Cf. A002129, A143336.

Sequence in context: A341834 A255275 A253104 * A143336 A328529 A053596

Adjacent sequences: A122855 A122856 A122857 * A122859 A122860 A122861

Keyword

sign

Author

Michael Somos, Sep 15 2006

Status

approved