OFFSET
0,4
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of f(q^3) * f(-q^4)^2 * chi(-q^6)^2 / chi(-q) in powers of q where chi(), f(), are Ramanujan theta functions.
Expansion of psi(q) * psi(q^2) * phi(-q^6)^2 / psi(-q^3) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of eta(q^2) * eta(q^4)^2 * eta(q^6)^5 / (eta(q) *eta(q^3) * eta(q^12)^3) in powers of q.
Euler transform of period 12 sequence [1, 0, 2, -2, 1, -4, 1, -2, 2, 0, 1, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (96 t)) = 18432^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A298420.
EXAMPLE
G.f. = 1 + q + q^2 + 3*q^3 + q^4 + 2*q^5 + q^6 - 2*q^7 - q^8 - 5*q^9 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ -q^3] QPochhammer[ q^4]^2 QPochhammer[ q^6, q^12]^2 QPochhammer[ -q, q], {q, 0, n}];
a[ n_] := SeriesCoefficient[ 2^(-3/2) EllipticTheta[ 2, 0, q^(1/2)] EllipticTheta[ 2, 0, q] EllipticTheta[ 4, 0, q^6]^2 / EllipticTheta[ 2, Pi/4, q^(3/2)], {q, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^4 + A)^2 * eta(x^6 + A)^5 / (eta(x + A) * eta(x^3 + A) * eta(x^12 + A)^3), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Jan 18 2018
STATUS
approved