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) * (psi(-q) * psi(q^4) / f(-q, -q^7)^2)^2 in powers of q where psi() is a Ramanujan theta function.
Euler transform of period 8 sequence [ 2, 0, -2, 0, -2, 0, 2, 0, ...].
Given g.f. A(x), then 0 = f(A(q), A(q^2)) where f(u, v) = (v - 1) * (u^2 - v) - 4*u*v.
Convolution inverse of A230535.
a(2*n) = 2 * A224216(n+1) unless n=0.
a(2*n) = 2 * A245432(n).
a(2*n - 1) = A245434(n).
a(4*n - 1) = A245433(n).
a(4*n + 2) = 2 * A210063(n).
G.f.: T(q) = 1/q + 2 + 3*q + ... for this sequence is quadratically related to T8E(q) of A029841: T8E = T + 2 + 1/T. - G. A. Edgar, Apr 12 2017
EXAMPLE
G.f. = 1/q + 2 + 3*q + 2*q^2 + q^3 - 2*q^4 - 4*q^5 - 4*q^6 + 6*q^8 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q^3, q^8] QPochhammer[ q^5, q^8] / (QPochhammer[ q^1, q^8] QPochhammer[ q^7, q^8]))^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, 0, 2, 0, 2, 0, -2][k%8 + 1]), n))};
(PARI) {a(n) = local(A, A2); if( n<-1, 0, n++; A = x * O(x^n); A = eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A)^3; A2 = subst(A, x, x^2); polcoeff( ((A^2 + A2) / (2 * A^2 * A2^2))^2, n))}
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Jul 21 2014
STATUS
approved