OFFSET
0,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of phi(q^3) * psi(-q)^3 / psi(-q^3) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of (eta(q) * eta(q^4) * eta(q^6)^2 / (eta(q^2) * eta(q^3) * eta(q^12)))^3 in powers of q.
Euler transform of period 12 sequence [-3, 0, 0, -3, -3, -3, -3, -3, 0, 0, -3, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 6^(3/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g(t) is g.f. for A245669.
a(3*n + 1) = -3 * A213056(n). a(6*n + 2) = 3 * A213592(n). a(6*n + 5) = 6 * A213607(n). a(8*n + 7) = 0.
Convolution cube of A089807.
EXAMPLE
G.f. = 1 - 3*q + 3*q^2 - q^3 - 3*q^4 + 6*q^5 - 3*q^6 + 3*q^8 + 3*q^9 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ EllipticTheta[3, Pi/3, q]^3, {q, 0, n}];
a[ n_] := SeriesCoefficient[ ((3 EllipticTheta[3, 0, q^9] - EllipticTheta[3, 0, q]) / 2)^3, {q, 0, n}];
a[ n_] := SeriesCoefficient[ (QPochhammer[-q^3, q^6] EllipticTheta[2, 0, Sqrt[-q]] / (2 (-q)^(1/8)))^3, {q, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)))^3, n))};
(Magma) A := Basis( ModularForms( Gamma0(12), 3/2), 67); A[1] - 3*A[2] + 3*A[3];
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Jul 28 2014
STATUS
approved