OFFSET
0,7
COMMENTS
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of q^(1/12) eta(q) / eta(q^3) in powers of q.
Euler transform of period 3 sequence [ -1, -1, 0, ...].
Given g.f. A(x) then B(q) = A(q^6)^2 / q satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = 4*v^2 + (u^2 - v) * (w^2 + v).
G.f. is a period 1 Fourier series which satisfies f(-1 / (432 t)) = 3^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A000726.
G.f.: Product_{k>0} (1 - x^(3*k-1)) * (1 - x^(3*k-2)).
Convolution inverse of A000726.
G.f.: (Product_{k>0} (1 + x^k + x^(2*k)))^-1. - Michael Somos, Oct 08 2015
a(n) = -(1/n)*Sum_{k=1..n} A046913(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017
EXAMPLE
G.f. = 1 - x - x^2 + x^3 - x^4 + 2*x^6 - x^7 - x^8 + 3*x^9 - 2*x^10 - x^11 + ...
G.f. = 1/q - q^11 - q^23 + q^35 - q^47 + 2*q^71 - q^83 - q^95 + 3*q^107 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x] / QPochhammer[ x^3], {x, 0, n}]; (* Michael Somos, Oct 08 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^3] QPochhammer[ x^2, x^3], {x, 0, n}]; (* Michael Somos, Oct 08 2015 *)
PROG
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) / eta(x^3 + A), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Jan 26 2008
STATUS
approved