A008658 - OEIS

A008658

Theta series of direct sum of 2 copies of D_4 lattice in powers of q^2.

1, 48, 624, 1344, 5232, 6048, 17472, 16512, 42096, 36336, 78624, 63936, 146496, 105504, 214656, 169344, 337008, 235872, 472368, 329280, 659232, 462336, 831168, 584064, 1178688, 756048, 1371552, 981120, 1799808, 1170720, 2201472

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

REFERENCES

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 119.

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 116, equ. (3) and p. 119, 10th equ.

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..10000

B. Brent, Quadratic Minima and Modular Forms, Experimental Mathematics, v.7 no.3 (1998), 257-274.

Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.]

J. McKay and A. Sebbar, Fuchsian groups, automorphic functions and Schwarzians, Math. Ann., 318 (2000), 255-275.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Index entries for sequences related to D_4 lattice

FORMULA

Fourier coefficients of E_{gamma,2}^2.

Convolution square of A004011. Convolution fourth power of A108096. - Michael Somos, Aug 20 2014

G.f.: (E_4(x) + 4*E_4(x^2)) / 5 where E_4() is the g.f. of A004009. [Ramanujan]. - Michael Somos, Feb 19 2017

Expansion of(2*phi(x)^4 - phi(-x)^4)^2 in powers of x where phi() is a Ramanujan theta function. - Michael Somos, Feb 19 2017

Expansion of phi(-x)^8 + 64*x * psi(x)^8 in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Feb 19 2017

Expansion of (phi(-x)^4 + 8*x * psi(x^2)^4)^2 in powers of x^2 where phi(), psi() are Ramanujan theta functions. - Michael Somos, Feb 19 2017

a(n) = 48*b(n) where b() is multiplicative with b(2^e) = 1 + 12*(8^e - 1) / 7, b(p^e) = (p^(3*(e+1)) - 1) / (p^3 - 1) if p>2. - Michael Somos, Feb 19 2017

EXAMPLE

G.f. = 1 + 48*x + 624*x^2 + 1344*x^3 + 5232*x^4 + 6048*x^5 + 17472*x^6 + ...

G.f. = 1 + 48*q^2+ 624*q^4 + 1344*q^6 + 5232*q^8 + 6048*q^10 + 17472*q^12 + ...

MATHEMATICA

a[ n_] := If[ n < 1, Boole[n == 0], 48 (DivisorSigma[3, n] + If[OddQ[n], 0, 4 DivisorSigma[3, n/2]])]; (* Michael Somos, Feb 19 2017 *)

PROG

(PARI) {a(n) = if( n<1, n==0, 48 * (sigma(n, 3) + if( n%2, 0, 4*sigma(n/2, 3))))}; /* Michael Somos, Jul 16 2004 */

(Magma) A := Basis( ModularForms( Gamma0(8), 4), 62); A[1] + 48*A[3] + 624*A[5]; /* Michael Somos, Aug 20 2014 */

CROSSREFS

Cf. A004009, A004011, A108096.

Sequence in context: A187611 A362235 A160286 * A215893 A136038 A233152

Adjacent sequences: A008655 A008656 A008657 * A008659 A008660 A008661