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A245436
Expansion of q^(-1) * (f(-q^3, -q^5) / f(-q, -q^7))^2 in powers of x where f(,) is Ramanujan's two-variable theta function.
3
1, 2, 3, 2, 1, -2, -4, -4, 0, 6, 9, 8, -1, -12, -20, -16, 1, 22, 38, 30, 1, -40, -64, -52, -2, 68, 107, 88, -2, -112, -180, -144, 3, 182, 292, 228, 4, -286, -452, -356, -4, 440, 686, 544, -5, -668, -1044, -816, 5, 996, 1563, 1210, 6, -1464, -2276, -1768, -8
OFFSET
-1,2
COMMENTS
Number 10 of the 15 generalized eta-quotients listed in Table I of Yang 2004.
A generator (Hauptmodul) of the function field associated with congruence subgroup Gamma_1(8). [Yang, 2004]
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
LINKS
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