A193522 Expansion of (1/q) * ((chi(q^3) * chi(-q^6)) / (chi(q) * chi(-q^2)))^4 in powers of q where chi() is a Ramanujan theta function.

1, -4, 14, -36, 85, -180, 360, -684, 1246, -2196, 3754, -6264, 10226, -16380, 25804, -40032, 61275, -92628, 138452, -204804, 300040, -435672, 627356, -896400, 1271525, -1791324, 2507426, -3488472, 4825531, -6638688, 9085888, -12373992

Offset

-1,2

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

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 - c(-q) * b(q^4) / (b(-q) * c(q^4)) in powers of q where b(), c() are cubic AGM functions.

Expansion of (eta(q) * eta(q^4)^2 * eta(q^6)^3 / (eta(q^2)^3 * eta(q^3) * eta(q^12)^2))^4 in powers of q.

Euler transform of period 12 sequence [ -4, 8, 0, 0, -4, 0, -4, 0, 0, 8, -4, 0, ...].

a(n) = -(-1)^n * A187091(n). a(2*n) = -4 * A128643(n).

a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n/3)) / (2 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2017

Example

1/q - 4 + 14*q - 36*q^2 + 85*q^3 - 180*q^4 + 360*q^5 - 684*q^6 + 1246*q^7 + ...

Mathematica

 QP := QPochhammer; A193522[n_]:= SeriesCoefficient[((QP[q]*QP[q^4]^2 *QP[q^6]^3)/(QP[q^2]^3*QP[q^3]*QP[q^12]^2))^4, {q, 0, n}]; Table[ A193522[n], {n, 0, 50}] (* G. C. Greubel, Dec 24 2017 *)

Prog

(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A)^2 * eta(x^6 + A)^3 / (eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^12 + A)^2))^4, n))}

Crossrefs

Cf. A128643, A187091.

Sequence in context: A079908 A038164 A327382 * A187091 A034528 A332834

Adjacent sequences: A193519 A193520 A193521 * A193523 A193524 A193525

Keyword

sign

Author

Michael Somos, Jul 29 2011

Status

approved