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A143336 Expansion of K(k) * (2 * E(k) - K(k)) / (Pi/2)^2 in powers of q where E(k), K(k) are complete elliptic integrals and q = exp(-Pi * K(k') / K(k)). 1
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, -248, -576, -560, -432 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

LINKS

Table of n, a(n) for n=0..53.

FORMULA

The generating function equals 0 when 2 * E(k) = K(k) at q = 0.1076539192... (A072558) the "One-Ninth" constant.

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

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

a(n) = (-1)^n * A122858(n). a(n) = -8 * A113184(n) unless n=0.

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_] := If[ n < 1, Boole[n == 0], -(-1)^n 8 Sum[(-1)^d d, {d, Divisors @ n}]]; (* Michael Somos, Apr 07 2015 *)

a[ n_] := SeriesCoefficient[ With[{m = InverseEllipticNomeQ[ q]}, EllipticK[ m] (2 EllipticE[ m] - EllipticK[ m]) (2/Pi)^2], {q, 0, n}]; (* Michael Somos, Apr 07 2015 *)

PROG

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

CROSSREFS

Cf. A113184, A122858.

Sequence in context: A255275 A253104 A122858 * A053596 A141384 A183400

Adjacent sequences:  A143333 A143334 A143335 * A143337 A143338 A143339

KEYWORD

sign

AUTHOR

Michael Somos, Aug 09 2008

STATUS

approved

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Last modified April 20 17:54 EDT 2019. Contains 322310 sequences. (Running on oeis4.)