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

1, -24, 24, -96, 24, -144, 96, -192, 24, -312, 144, -288, 96, -336, 192, -576, 24, -432, 312, -480, 144, -768, 288, -576, 96, -744, 336, -960, 192, -720, 576, -768, 24, -1152, 432, -1152, 312, -912, 480, -1344, 144, -1008, 768, -1056, 288, -1872

Offset: 0

Author

Ralf Stephan, Feb 18 2005

Keywords

• sign
• look

Comments

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

Examples

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000
• C. Pache, Shells of selfdual lattices viewed as spherical designs
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Eisenstein Series
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A004011.

Programs

Magma

A := Basis( ModularForms( Gamma0(4), 2), 46); A[1] - 24*A[2]; /* Michael Somos, Aug 21 2014 */

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^4 - EllipticTheta[ 2, 0, q]^4, {q, 0, n}]; (* Michael Somos, Jun 01 2012 *)
a[ n_] := With[{m = InverseEllipticNomeQ @x}, SeriesCoefficient[(1 - 2 m) (EllipticK[m] / (Pi/2))^2, {x, 0, n}]]; (* Michael Somos, Aug 21 2014 *)
a[ n_] := If[ n < 1, Boole[n == 0], -24 Sum[ (-1)^(n + d) n / d, { d, Divisors[ n]}]]; (* Michael Somos, Aug 21 2014 *)
a[ n_] := If[ n < 1, Boole[n == 0], -24 DivisorSum[ n, (-1)^(n + #) n / # &]]; (* Michael Somos, Aug 21 2014 *)

PARI

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

Formula

G.f.: theta_4(q)^4 - theta_2(q)^4. - Michael Somos, May 29 2005

a(n) = (-1)^n * A004011(n). - Michael Somos, Jun 01 2012

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