OFFSET
1,2
COMMENTS
LINKS
Seiichi Manyama, Table of n, a(n) for n = 1..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of (eta(q^2) / (eta(q) * eta(q^4)))^8 in powers of q.
a(n) is multiplicative with a(2) = 8, a(2^e) = -(-8)^e if e>1, a(p^e) = a(p) * a(p^(e-1)) - p^7 * a(p^(e-2)) if p>2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 256 (t/i)^8 f(t) where q = exp(2 Pi i t).
EXAMPLE
G.f. = q + 8*q^2 + 12*q^3 - 64*q^4 - 210*q^5 + 96*q^6 + 1016*q^7 + 512*q^8 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q^2] EllipticTheta[ 2, 0, q^(1/2)] / 2)^8, {q, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( ( eta(x^2 + A)^4 / eta(x + A) / eta(x^4 + A) )^8, n))};
(Magma) A := Basis( CuspForms( Gamma0(4), 8), 39); A[1] + 8*A[2]; /* Michael Somos, Jun 10 2015 */
CROSSREFS
KEYWORD
AUTHOR
Michael Somos, Apr 10 2013
STATUS
approved