A264026 Expansion of (f(x^3) / f(x))^6 in powers of x where f() is a Ramanujan theta function.

1, -6, 27, -92, 279, -756, 1913, -4536, 10260, -22220, 46479, -94176, 185749, -357426, 673056, -1242404, 2252772, -4017816, 7058609, -12228060, 20911230, -35330324, 59023728, -97568712, 159693831, -258941124, 416181510, -663337512, 1048935414, -1646245836

Offset

0,2

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 = 0..2500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1/2) * (eta(q) * eta(q^4) * eta(q^6)^3 / (eta(q^2)^3 * eta(q^3) * eta(q^12)))^6 in powers of q.

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

G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = (1/27) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132107.

Convolution inverse of A132107.

a(n) ~ (-1)^n * exp(2*Pi*sqrt(2*n/3)) / (2^(3/4) * 3^(13/4) * n^(3/4)). - Vaclav Kotesovec, Jun 06 2018

Example

G.f. = 1 - 6*x + 27*x^2 - 92*x^3 + 279*x^4 - 756*x^5 + 1913*x^6 - 4536*x^7 + ...
G.f. = q - 6*q^3 + 27*q^5 - 92*q^7 + 279*q^9 - 756*q^11 + 1913*q^13 - 4536*q^15 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ -x^3] / QPochhammer[ -x])^6, {x, 0, n}];
eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q^(-1/2)* (eta[q]*eta[q^4]*eta[q^6]^3/(eta[q^2]^3*eta[q^3]*eta[q^12]))^6, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jul 04 2018 *)

Prog

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

Crossrefs

Cf. A000726, A132107.

Sequence in context: A038166 A327384 A121596 * A341385 A344100 A136747

Adjacent sequences: A264023 A264024 A264025 * A264027 A264028 A264029

Keyword

sign

Author

Michael Somos, Nov 01 2015

Status

approved