OFFSET
1,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q * (chi(q) * psi(-q^3))^3 in powers of q where chi(), psi() are Ramanujan theta functions.
Expansion of (eta(q^2)^2 * eta(q^3) * eta(q^12) / (eta(q) * eta(q^4) * eta(q^6)))^3 in powers of q.
Euler transform of period 12 sequence [ 3, -3, 0, 0, 3, -3, 3, 0, 0, -3, 3, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 2^(3/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A245668.
G.f.: x * (Sum_{k in Z} x^(3*k^2 + 2*k))^3.
Convolution cube of A089801.
a(2*n + 2) = A005885(n).
EXAMPLE
G.f. = q + 3*q^2 + 3*q^3 + q^4 + 3*q^6 + 6*q^7 + 3*q^8 + 3*q^9 + 6*q^10 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ ((EllipticTheta[ 3, 0, q^(1/3)] - EllipticTheta[ 3, 0, q^3]) / 2)^3, {q, 0, n}];
a[ n_] := SeriesCoefficient[ (QPochhammer[ -q, q^2] EllipticTheta[ 2, Pi/4, q^(3/2)])^3 / (2^(3/2) q^(1/8)), {q, 0, n}];
PROG
(PARI) {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)))^3, n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Jul 28 2014
STATUS
approved