OFFSET
0,8
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^(1/4) * eta(q) / eta(q^7) in powers of q.
Euler transform of period 7 sequence [ -1, -1, -1, -1, -1, -1, 0, ...].
Given g.f. A(x), then B(q) = A(q^4) / q satisifes 0 = f(B(q), B(q^3)) where f(u, v) = (u - v^3) * (u^3 - v) - 3*u*v * (2 + u*v).
G.f. is a period 1 Fourier series which satisfies f(-1 / (112 t)) = 7^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A035985.
G.f.: Product_{k>0} (1 - x^k) / (1 - x^(7*k)).
Convolution inverse of A035985.
a(7*n + 3) = a(7*n + 4) = a(7*n + 6) = 0.
a(n) = -(1/n)*Sum_{k=1..n} A113957(k)*a(n-k), a(0) = 1. - Seiichi Manyama, Mar 25 2017
EXAMPLE
G.f. = 1 - x - x^2 + x^5 + 2*x^7 - x^8 - x^9 + 3*x^14 - 3*x^15 - 2*x^16 + ...
G.f. = q^-1 - q^3 - q^7 + q^19 + 2*q^27 - q^31 - q^35 + 3*q^55 - 3*q^59 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x] / QPochhammer[ x^7], {x, 0, n}];
PROG
(PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) / eta(x^7 + A), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Jul 01 2014
STATUS
approved