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.

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

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

Cf. A112143, A224216, A230535, A245432, A245433, A245434.

Sequence in context: A340680 A343897 A193585 * A285581 A222173 A301368

Adjacent sequences: A245433 A245434 A245435 * A245437 A245438 A245439

Keyword

sign

Author

Michael Somos, Jul 21 2014

Status

approved