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

0,2

Comments

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

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

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

Example

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

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 *)

Prog

(PARI) {a(n) = if( n<1, n==0, (-1)^n * 24 * sumdiv(n, d, d%2*d))}; /* Michael Somos, May 29 2005 */
(Magma) A := Basis( ModularForms( Gamma0(4), 2), 46); A[1] - 24*A[2]; /* Michael Somos, Aug 21 2014 */

Crossrefs

Cf. A004011.

Sequence in context: A183008 A022358 A122505 * A004011 A334570 A056465

Adjacent sequences: A103637 A103638 A103639 * A103641 A103642 A103643

Keyword

sign,look

Author

Ralf Stephan, Feb 18 2005

Status

approved