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A103640 Expansion of theta_4(q)^4 - theta_2(q)^4, where theta_2 and theta_4 are the Jacobi theta series. 2

%I

%S 1,-24,24,-96,24,-144,96,-192,24,-312,144,-288,96,-336,192,-576,24,

%T -432,312,-480,144,-768,288,-576,96,-744,336,-960,192,-720,576,-768,

%U 24,-1152,432,-1152,312,-912,480,-1344,144,-1008,768,-1056,288,-1872

%N Expansion of theta_4(q)^4 - theta_2(q)^4, where theta_2 and theta_4 are the Jacobi theta series.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%H Seiichi Manyama, <a href="/A103640/b103640.txt">Table of n, a(n) for n = 0..10000</a>

%H C. Pache, <a href="http://arXiv.org/abs/math.CO/0502313">Shells of selfdual lattices viewed as spherical designs</a>

%H M. Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EisensteinSeries.html">Eisenstein Series</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F G.f.: theta_4(q)^4 - theta_2(q)^4. - _Michael Somos_, May 29 2005

%F a(n) = (-1)^n * A004011(n). - _Michael Somos_, Jun 01 2012

%F Expansion of phi(-q)^4 - 16*q*psi(q^2)^4 in powers of q where phi(), psi() are Ramanujan theta functions. - _Michael Somos_, Aug 21 2014

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

%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^4 - EllipticTheta[ 2, 0, q]^4, {q, 0, n}]; (* _Michael Somos_, Jun 01 2012 *)

%t a[ n_] := With[{m = InverseEllipticNomeQ @x}, SeriesCoefficient[(1 - 2 m) (EllipticK[m] / (Pi/2))^2, {x, 0, n}]]; (* _Michael Somos_, Aug 21 2014 *)

%t a[ n_] := If[ n < 1, Boole[n == 0], -24 Sum[ (-1)^(n + d) n / d, { d, Divisors[ n]}]]; (* _Michael Somos_, Aug 21 2014 *)

%t a[ n_] := If[ n < 1, Boole[n == 0], -24 DivisorSum[ n, (-1)^(n + #) n / # &]]; (* _Michael Somos_, Aug 21 2014 *)

%o (PARI) {a(n) = if( n<1, n==0, (-1)^n * 24 * sumdiv(n, d, d%2*d))}; /* _Michael Somos_, May 29 2005 */

%o (MAGMA) A := Basis( ModularForms( Gamma0(4), 2), 46); A[1] - 24*A[2]; /* _Michael Somos_, Aug 21 2014 */

%Y Cf. A004011.

%K sign,look

%O 0,2

%A _Ralf Stephan_, Feb 18 2005

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Last modified April 1 02:00 EDT 2020. Contains 333153 sequences. (Running on oeis4.)