OFFSET
0,4
COMMENTS
REFERENCES
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
G. Nebe and N. J. A. Sloane, Home page for lattice
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
theta_3(z)^24 - 48*theta_3(z)^16*Del8(z) + 48*theta_3(z)^8*Del8(z)^2 where Del8(z) = [(theta_2(z)*theta_4(z))^4]/16 is the unique cusp form of weight 8 for G(2)... SPLAG p. 187.
Expansion of u * (u^2 - 48*u*v + 48*v^2) in powers of q where u = phi(q)^8, v = q * psi(-q)^8 and phi(), psi() are Ramanujan theta functions. - Michael Somos, Feb 01 2017
Expansion of 16*q * f(q)^8 * (u^4 + u^3*v + 3/16*u^2*v^2 + u*v^3 + v^4) / (u * v) in powers of q where u = f(-q)^8, v = 16*q * f(-q^4)^8 and f() is a Ramanujan theta function. - Michael Somos, Feb 01 2017
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 4096 (t/i)^12 f(t) where q = exp(2 Pi i t). - Michael Somos, Feb 01 2017
EXAMPLE
G.f. = 1 + 4096*q^3 + 98256*q^4 + 1130496*q^5 + 8384512*q^6 + ...
MATHEMATICA
a[ n_] := With[ {u = EllipticTheta[ 3, 0, q]^8, v = EllipticTheta[ 2, Pi/4, q^(1/2)]^8}, SeriesCoefficient[ u (u^2 - 3 u v + 3/16 v^2), {q, 0, n}]]; (* Michael Somos, Feb 01 2017 *)
PROG
(PARI) {a(n) = if( n<0, 0, my(A, u, v); A = x * O(x^n); u = eta(x^2 + A)^40 / (eta(x + A) * eta(x^4 + A))^16; v = x * (eta(x + A) * eta(x^4 + A))^8 / eta(x^2 + A)^8; polcoeff( u * (u^2 - 48*u*v + 48*v^2), n))}; /* Michael Somos, Feb 01 2017 */
(PARI) {a(n) = if( n<0, 0, my(A, u, v); A = x * O(x^n); u = eta(x + A)^8; v = 16*x * eta(x^4 + A)^8; polcoeff( 16*x * eta(-x + A)^8 * (u^4 + u^3*v + 3/16*u^2*v^2 + u*v^3 + v^4) / (u * v), n))}; /* Michael Somos, Feb 01 2017 */
(Magma) A := Basis( ModularForms( Gamma0(4), 12), 23); A[1] + 4096*A[4] + 98256*A[5] + 1130496*A[6] + 8384512*A[7]; /* Michael Somos, Feb 01 2017 */
CROSSREFS
KEYWORD
nonn,easy,nice
AUTHOR
EXTENSIONS
More terms from Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), May 16 2000
STATUS
approved