OFFSET
0,2
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 phi(-x^2) * phi(-x^3) * phi(-x^6) / chi(-x)^3 in powers of x where phi(), chi() are Ramanujan theta functions.
Expansion of q^(-1/4) * eta(q^2)^5 * eta(q^3)^2 * eta(q^6) / (eta(q)^3 * eta(q^4) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [3, -2, 1, -1, 3, -5, 3, -1, 1, -2, 3, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (96 t)) = 48^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A298421.
EXAMPLE
G.f. = 1 + 3*x + 4*x^2 + 5*x^3 + 6*x^4 + 8*x^5 + 9*x^6 + 6*x^7 + 8*x^8 + ...
G.f. = q + 3*q^9 + 4*q^17 + 5*q^25 + 6*q^33 + 8*q^41 + 9*q^49 + 6*q^57 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ -q] QPochhammer[ -q, q]^2 QPochhammer[ q^3]^2 QPochhammer[ q^6, q^12], {q, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^2] EllipticTheta[ 4, 0, x^3] EllipticTheta[ 4, 0, x^6] QPochhammer[ -x, x]^3, {x, 0, n}];
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^2 * eta(x^6 + A) / (eta(x + A)^3 * eta(x^4 + A) * eta(x^12 + A)), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Jan 18 2018
STATUS
approved