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Expansion of K(k) * (6 * E(k) - (1 + 4*k'^2) * 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 #6 Apr 07 2015 21:06:54

%S 1,24,-72,96,-168,144,-288,192,-360,312,-432,288,-672,336,-576,576,

%T -744,432,-936,480,-1008,768,-864,576,-1440,744,-1008,960,-1344,720,

%U -1728,768,-1512,1152,-1296,1152,-2184,912,-1440,1344,-2160,1008,-2304,1056,-2016,1872,-1728,1152

%N Expansion of K(k) * (6 * E(k) - (1 + 4*k'^2) * 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 G.f.: 1 - 24 * Sum_{k>0} k * (-x)^k / (1 - (-x)^k) = 1 - 24 * Sum_{k>0} (-x)^k / (1 - (-x)^k)^2.

%F a(n) = (-1)^n * A006352(n).

%F a(n) = 24 * A143348(n) unless n=0.

%F Expansion of -(P(q) - 6 * P(q^2) + 4 * P(q^4)) in powers of q where P() is a Ramanujan Lambert series. - _Michael Somos_, Apr 07 2015

%e G.f. = 1 + 24*q - 72*q^2 + 96*q^3 - 168*q^4 + 144*q^5 - 288*q^6 + 192*q^7 + ...

%t a[ n_] := If[ n < 1, Boole[n == 0], -(-1)^n 24 DivisorSigma[ 1, n]]; (* _Michael Somos_, Apr 07 2015 *)

%o (PARI) {a(n) = if( n<1, n==0, -(-1)^n * 24 * sigma(n))};

%Y Cf. A006352, A143348.

%K sign

%O 0,2

%A _Michael Somos_, Aug 09 2008