A122858 - OEIS

%I #39 Jul 07 2020 17:58:09

%S 1,8,-8,32,-40,48,-32,64,-104,104,-48,96,-160,112,-64,192,-232,144,

%T -104,160,-240,256,-96,192,-416,248,-112,320,-320,240,-192,256,-488,

%U 384,-144,384,-520,304,-160,448,-624,336,-256,352,-480,624,-192,384,-928,456

%N 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)).

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

%H G. C. Greubel, <a href="/A122858/b122858.txt">Table of n, a(n) for n = 0..10000</a>

%H A. Brini and A. Tanzini, <a href="http://arxiv.org/abs/0804.2598">Exact results for topological strings on resolved Y(p,q) singularities</a>, arXiv:0804.2598 [hep-th], 2008-2009, p. 40, equation (6.82).

%F Expansion of (2 * E(k) - k'^2 * K(k)) * K(k) * (2/Pi)^2 in powers of q.

%F Expansion of (E(k) + k' * K(k)) * K(k) * (2/Pi)^2 / 2 in powers of q^4.

%F Expansion of (4 * P(q^2) - P(q)) / 3 in powers of q where P() is a Ramanujan Lambert series.

%F G.f.: 1 + 8 * Sum_{k>0} x^k / (1 + x^k)^2.

%F G.f.: 1 - 8 * Sum_{k>0} k * (-x)^k / (1 - x^k).

%F G.f.: 1 + 8 * Sum_{k>0} k * x^k * (1 - 3*x^k) / (1 - x^(2*k)).

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

%F 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

%F Conjecture: -3 A122858(n) - A229616(n) + 4 A282031(n) = 0 for all n. - _Thomas Baruchel_, Jun 23 2018

%e 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 + ...

%t 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 *)

%t a[ n_] := If[ n < 1, Boole[n == 0], -8 DivisorSum[ n, # (-1)^# &]]; (* _Michael Somos_, Jun 02 2015 *)

%t 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 *)

%t CoefficientList[Series[(2/Pi) EllipticE[InverseEllipticNomeQ[Sqrt[q]]] EllipticTheta[3, 0, Sqrt[q]]^2, {q, 0, 40}], q] (* _Jan Mangaldan_, Jul 07 2020 *)

%o (PARI) {a(n) = if( n<1, n==0, -8 * sumdiv(n, d, (-1)^d * d))};

%Y Cf. A002129, A143336.

%K sign

%O 0,2

%A _Michael Somos_, Sep 15 2006