OFFSET
0,2
COMMENTS
LINKS
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of eta(q^2)^9 / (eta(q)^2 * eta(q^4)^4) in powers of q.
Expansion of phi(q) * phi(-q^2)^2 = phi(-q^2)^4 / phi(-q) in powers of q.
Euler transform of period 4 sequence [2, -7, 2, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2^(11/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A045834.
G.f. Product_{k>0} (1 - x^k)^3 * (1 + x^k)^5 / (1 + x^(2*k))^4.
EXAMPLE
G.f. = 1 + 2*x - 4*x^2 - 8*x^3 + 6*x^4 + 8*x^5 - 8*x^6 + 12*x^8 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q]^2, {q, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q^2]^2, {q, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^9 / (eta(x + A)^2 * eta(x^4 + A)^4), n))};
(Magma) A := Basis( ModularForms( Gamma0(16), 3/2), 66); A[1] + 2*A[2] - 4*A[3] - 8*A[4];
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Sep 09 2018
STATUS
approved