OFFSET
-1,3
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^(-1) * f(-q^4, -q^5)^3 * f(-q^3) / (f(-q) * f(-q^9)^3) in powers of q where f() is a Ramanujan theta function.
Euler transform of period 9 sequence [ 1, 1, 0, -2, -2, 0, 1, 1, 0, ...].
Given g.f. A(q), then 0 = f(A(q), A(q^2)) where f(u, v) = (u^2 - v) * (u*v^2 + v^2 + 1) - 2 * (u + v + u*v)^2.
Given g.f. A(q), then 0 = f(A(q), A(q^3)) where f(u, v) = (v^2 + v + 1) * (u^3 - v) - 3*u*v * (u + 1) * (v + 2).
a(n) = A245421(n) unless n=0.
EXAMPLE
G.f. = 1/q + 1 + 2*q + 2*q^2 + q^3 - q^4 - 2*q^5 - 3*q^6 - 2*q^7 + q^8 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q^4, q^9] QPochhammer[ q^5, q^9])^3 QPochhammer[ q^3] / (QPochhammer[ q] ), {q, 0, n}];
a[ n_] := If[ n < -1, 0, With[{m = n + 1}, SeriesCoefficient[ 1/q Product[ (1 - q^k)^{-1, -1, 0, 2, 2, 0, -1, -1, 0}[[Mod[k, 9, 1]]], {k, m}], {q, 0, n}]]];
a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q^4, q^9] QPochhammer[ q^5, q^9])^2 / (QPochhammer[ q^1, q^9] QPochhammer[ q^2, q^9] QPochhammer[ q^7, q^9] QPochhammer[ q^8, q^9] ), {q, 0, n}];
PROG
(PARI) {a(n) = if( n<-1, 0, n++; polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[0, -1, -1, 0, 2, 2, 0, -1, -1][k%9 + 1]), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Jul 21 2014
STATUS
approved