cp's OEIS Frontend

A008658 Theta series of direct sum of 2 copies of D_4 lattice in powers of q^2.

%I A008658 #42 Feb 16 2025 08:32:32
%S A008658 1,48,624,1344,5232,6048,17472,16512,42096,36336,78624,63936,146496,
%T A008658 105504,214656,169344,337008,235872,472368,329280,659232,462336,
%U A008658 831168,584064,1178688,756048,1371552,981120,1799808,1170720,2201472
%N A008658 Theta series of direct sum of 2 copies of D_4 lattice in powers of q^2.
%C A008658 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%D A008658 J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 119.
%D A008658 Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 116, equ.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 A008658 Seiichi Manyama, <a href="/A008658/b008658.txt">Table of n, a(n) for n = 0..10000</a>
%H A008658 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 A008658 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, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021.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 A008658 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 A008658 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A008658 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A008658 <a href="/index/Da#D4">Index entries for sequences related to D_4 lattice</a>
%F A008658 Fourier coefficients of E_{gamma,2}^2.
%F A008658 Convolution square of A004011. Convolution fourth power of A108096. - _Michael Somos_, Aug 20 2014
%F A008658 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 A008658 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 A008658 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 A008658 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 A008658 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 A008658 G.f. = 1 + 48*x + 624*x^2 + 1344*x^3 + 5232*x^4 + 6048*x^5 + 17472*x^6 + ...
%e A008658 G.f. = 1 + 48*q^2+ 624*q^4 + 1344*q^6 + 5232*q^8 + 6048*q^10 + 17472*q^12 + ...
%t A008658 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 A008658 (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 A008658 (Magma) A := Basis( ModularForms( Gamma0(8), 4), 62); A[1] + 48*A[3] + 624*A[5]; /* _Michael Somos_, Aug 20 2014 */
%Y A008658 Cf. A004009, A004011, A108096.
%K A008658 nonn,easy
%O A008658 0,2
%A A008658 _N. J. A. Sloane_
%E A008658 Additional comments from Barry Brent (barryb(AT)primenet.com)