A193557 Expansion of (1/q) * chi(-q) * chi(-q^3) * chi(-q^6)^4 / chi(q)^4 in powers of q where chi() is a Ramanujan theta function.

1, -5, 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 - (b(q^2) * c(q^2))^3 / (b(-q)^2 * c(-q) * b(q^4) * c(q^4)^2) in powers of q where b(), c() are cubic AGM functions.

Expansion of eta(q)^5 * eta(q^3) * eta(q^4)^4 * eta(q^6)^3 / (eta(q^2)^9 * eta(q^12)^4) in powers of q.

Euler transform of period 12 sequence [ -5, 4, -6, 0, -5, 0, -5, 0, -6, 4, -5, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (1+u)^2 * v^4 - u^4 * v^2 * (1+v) - 4*u^2 * (1+u) * (1+v) *(4+v) * (4+3*v).

a(n) = -(-1)^n * A187198(n). a(n) = A193522(n) unless n=0. a(2*n) = -4 * A128643(n) unless n=0.

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

Mathematica

eta[q_]:= q^(1/24)*QPochhammer[q]; a[n_] := SeriesCoefficient[eta[q]^5* eta[q^3]*eta[q^4]^4*eta[q^6]^3/(eta[q^2]^9*eta[q^12]^4), {q, 0, n}]; Table[a[n], {n, -1, 50}] (* G. C. Greubel, Apr 03 2018 *)

Prog

(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A)^5 * eta(x^3 + A) * eta(x^4 + A)^4 * eta(x^6 + A)^3 / (eta(x^2 + A)^9 * eta(x^12 + A)^4), n))}

Crossrefs

Cf. A128643, A187198, A193522.

Sequence in context: A211562 A261055 A320853 * A187198 A097507 A052951

Adjacent sequences: A193554 A193555 A193556 * A193558 A193559 A193560

Keyword

sign

Author

Michael Somos, Jul 30 2011

Status

approved