A008658 - OEIS

%I #41 Sep 08 2022 08:44:36

%S 1,48,624,1344,5232,6048,17472,16512,42096,36336,78624,63936,146496,

%T 105504,214656,169344,337008,235872,472368,329280,659232,462336,

%U 831168,584064,1178688,756048,1371552,981120,1799808,1170720,2201472

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

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

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

%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 116, equ.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

%H Seiichi Manyama, <a href="/A008658/b008658.txt">Table of n, a(n) for n = 0..10000</a>

%H B. Brent, <a href="http://projecteuclid.org/euclid.em/1047674207">Quadratic Minima and Modular Forms</a>, Experimental Mathematics, v.7 no.3 (1998), 257-274.

%H Masao Koike, <a href="https://oeis.org/A004016/a004016.pdf">Modular forms on non-compact arithmetic triangle groups</a>, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021.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.]

%H J. McKay and A. Sebbar, <a href="http://dx.doi.org/10.1007/s002080000116">Fuchsian groups, automorphic functions and Schwarzians</a>, Math. Ann., 318 (2000), 255-275.

%H Michael 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/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%H <a href="/index/Da#D4">Index entries for sequences related to D_4 lattice</a>

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

%F Convolution square of A004011. Convolution fourth power of A108096. - _Michael Somos_, Aug 20 2014

%F 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

%F 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

%F 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

%F 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

%F 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

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

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

%t 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 *)

%o (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 */

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

%Y Cf. A004009, A004011, A108096.

%K nonn,easy

%O 0,2

%A _N. J. A. Sloane_

%E Additional comments from Barry Brent (barryb(AT)primenet.com)