login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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)). 4
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 (list; graph; refs; listen; history; text; internal format)
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, 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 *)

PROG

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

CROSSREFS

Cf. A002129, A143336.

Sequence in context: A133038 A255275 A253104 * A143336 A328529 A053596

Adjacent sequences:  A122855 A122856 A122857 * A122859 A122860 A122861

KEYWORD

sign

AUTHOR

Michael Somos, Sep 15 2006

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 21 20:44 EDT 2019. Contains 328315 sequences. (Running on oeis4.)