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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

%I

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

%T -232,-144,-104,-160,-240,-256,-96,-192,-416,-248,-112,-320,-320,-240,

%U -192,-256,-488,-384,-144,-384,-520,-304,-160,-448,-624,-336,-256,-352,-480,-624,-192,-384,-928,-456,-248,-576,-560,-432

%N 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)).

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

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

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

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

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

%t 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 *)

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

%Y Cf. A113184, A122858.

%K sign

%O 0,2

%A _Michael Somos_, Aug 09 2008

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Last modified May 25 16:08 EDT 2019. Contains 323572 sequences. (Running on oeis4.)