%I #29 Sep 08 2022 08:44:35
%S 1,0,12,64,60,0,160,384,252,0,312,960,544,0,960,1664,1020,0,876,2880,
%T 1560,0,2400,4224,2080,0,2040,5248,3264,0,4160,7680,4092,0,3480,9984,
%U 4380,0,7200,10880,6552,0,4608,14784,8160,0,10560,17664,8224,0,7812,18560
%N Theta series of {D_6}* lattice.
%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 New York, 1999, ISBN 978-1-4757-6568-7, p. 120.
%H G. C. Greubel, <a href="/A008425/b008425.txt">Table of n, a(n) for n = 0..10000</a>
%H G. Nebe and N. J. A. Sloane, <a href="http://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Ds6.html">Home page for this lattice</a>
%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>
%F Apparently, a(n) = Sum_{d|n} d^2*(4*(Kronecker(-4,n/d) - Kronecker(-4,d))), n > 0. - _Ralf Stephan_, Dec 31 2014
%F Expansion of phi(x^2)^6 + 64 * x^3 * psi(x^4)^6 in powers of x where phi(), psi() are Ramanujan theta functions. - _Michael Somos_, Dec 14 2016
%F G.f. is a period 1 Fourier series that satisfies f(-1 / (8 t)) = 16 (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A008428. - _Michael Somos_, Dec 14 2016
%F G.f.: theta_3(0, x^2)^6 + theta_2(0, x^2)^6.
%e G.f. = 1 + 12*x^2 + 64*x^3 + 60*x^4 + 160*x^6 + 384*x^7 + 252*x^8 + 312*x^10 + ...
%t a[n_] := DivisorSum[n, #^2*(4*(KroneckerSymbol[-4, n/#]-KroneckerSymbol[-4, #]))&]; a[0]=1; Table[a[n], {n, 0, 40}] (* _Jean-François Alcover_, Oct 06 2016, after _Ralf Stephan_ *)
%t a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^2]^6 + EllipticTheta[ 2, 0, x^2]^6, {x, 0, n}]; (* _Michael Somos_, Dec 14 2016 *)
%o (PARI) {a(n) = if( n<1, n==0, 4 * sumdiv(n, d, d^2 * (kronecker(-4, n/d) - kronecker(-4, d))))}; /* _Michael Somos_, Dec 14 2016 */
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^4 + A)^5 / (eta(x^2 + A) * eta(x^8 + A))^2)^6 + 64 * x^3 * (eta(x^8 + A)^2/ eta(x^4 + A))^6, n))}; /* _Michael Somos_, Dec 14 2016 */
%o (PARI) {a(n) = my(G); if( n<0, 0, G = [2, 0, 0, 0, 0, 1; 0, 2, 0, 0, 0, 1; 0, 0, 2, 0, 0, 1; 0, 0, 0, 2, 0, 1; 0, 0, 0, 0, 2, 1; 1, 1, 1, 1, 1, 3]; polcoeff( 1 + 2 * x * Ser( qfrep( G, n)), n))}; /* _Michael Somos_, Dec 14 2016 */
%o (Magma) A := Basis( ModularForms( Gamma1(8), 3), 52); A[1] + 12*A[3] + 64*A[4] + 60*A[5] + 160*A[7]; /* _Michael Somos_, Dec 14 2016 */
%Y Cf. A000141, A008428.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_