OFFSET
0,2
COMMENTS
REFERENCES
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 118.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
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
G.f.: (theta_3(q^(1/2))^6 + theta_4(q^(1/2))^6)/2
Expansion of ( phi(q)^6 + phi(-q)^6 ) / 2 in powers of q^2 where phi() is a Ramanujan theta function. - Michael Somos, Sep 14 2007
a(n) = A000141(2*n).
G.f. is a period 1 Fourier series that satisfies f(-1 / (8 t)) = 12 (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A008425. - Michael Somos, Aug 26 2015
EXAMPLE
G.f. = 1 + 60*x + 252*x^2 + 544*x^3 + 1020*x^4 + 1560*x^5 + 2080*x^6 + ...
G.f. = 1 + 60*q^2 + 252*q^4 + 544*q^6 + 1020*q^8 + 1560*q^10 + 2080*q^12 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x]^6, {x, 0, 2 n}]; (* Michael Somos, Aug 26 2015 *)
PROG
(PARI) {a(n) = if( n<1, n==0, 4 * sumdiv(n, d, d^2 * (16 * kronecker(-4, n/d) - kronecker(-4, d))))}; /* Michael Somos, Nov 03 2006 */
(PARI) {a(n) = if( n<0, 0, n*=2; polcoeff( sum(k=1, sqrtint(n), 2*x^k^2, 1 + x * O(x^n))^6, n))}; /* Michael Somos, Nov 03 2006 */
(Magma) A := Basis( ModularForms( Gamma1(8), 3), 80); A[1] + 60*A[3] + 252*A[5] + 544*A[7]; /* Michael Somos, Aug 26 2015 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved