OFFSET
0,3
COMMENTS
REFERENCES
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.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..10000
G. Nebe and N. J. A. Sloane, Home page for this lattice
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Apparently, a(n) = Sum_{d|n} d^2*(4*(Kronecker(-4,n/d) - Kronecker(-4,d))), n > 0. - Ralf Stephan, Dec 31 2014
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
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
G.f.: theta_3(0, x^2)^6 + theta_2(0, x^2)^6.
EXAMPLE
G.f. = 1 + 12*x^2 + 64*x^3 + 60*x^4 + 160*x^6 + 384*x^7 + 252*x^8 + 312*x^10 + ...
MATHEMATICA
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 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^2]^6 + EllipticTheta[ 2, 0, x^2]^6, {x, 0, n}]; (* Michael Somos, Dec 14 2016 *)
PROG
(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 */
(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 */
(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 */
(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 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved