OFFSET
-1,2
COMMENTS
LINKS
G. A. Edgar, Table of n, a(n) for n = -1..1003
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Y. Yang, Transformation formulas for generalized Dedekind eta functions, Bull. London Math. Soc. 36 (2004), no. 5, 671-682. See p. 679, Table 1.
FORMULA
Expansion of q^(-1) * f(-q) * f(-q^9)^3 / (f(-q^3) * f(-q, -q^8)^3) in powers of q where f() is a Ramanujan theta function.
Euler transform of period 9 sequence [ 2, -1, 0, -1, -1, 0, -1, 2, 0, ...].
Given g.f. A(q), then 0 = f(A(q), A(q^2)) where f(u, v) = (u^2 + v) * (u*v^2 + u - 1) - 2*u*v * (u + v - 1)^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) * (2*v - 1);
a(n) = A245424(n) unless n=0.
G.f. T(q) = 1/q + 2 + 2*q + ... for this function is cubically related to T9B(q) of A058091: T9B = T - 2 - 1/T - 1/(T-1). - G. A. Edgar, Apr 13 2017
EXAMPLE
G.f. = 1/q + 2 + 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] / (QPochhammer[ q^3] (QPochhammer[ q^1, q^9] QPochhammer[ q^8, q^9])^3), {q, 0, n}];
a[ n_] := If[ n < -1, 0, With[{m = n + 1}, SeriesCoefficient[ 1/q Product[ (1 - q^k)^{-2, 1, 0, 1, 1, 0, 1, -2, 0}[[Mod[k, 9, 1]]], {k, m}], {q, 0, n}]]];
a[ n_] := SeriesCoefficient[ 1/q QPochhammer[ q^2, q^9] QPochhammer[ q^4, q^9] QPochhammer[ q^5, q^9] QPochhammer[ q^7, q^9] / (QPochhammer[ q^1, q^9] QPochhammer[ q^8, q^9])^2, {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, -2, 1, 0, 1, 1, 0, 1, -2][k%9 + 1]), n))};
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Jul 21 2014
STATUS
approved