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
Offset: 0
Examples
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 + ...
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- Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 116, equ.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- Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021.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
Programs
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Magma
A := Basis( ModularForms( Gamma0(8), 4), 62); A[1] + 48*A[3] + 624*A[5]; /* Michael Somos, Aug 20 2014 */
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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 *)
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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 */
Formula
Fourier coefficients of E_{gamma,2}^2.
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
Extensions
Additional comments from Barry Brent (barryb(AT)primenet.com)
Comments