A133078 Expansion of s(q)^4 in powers of q where s() is a cubic AGM function.

1, -12, 54, -84, -147, 756, -756, -1212, 3510, -2028, -3402, 7992, -6132, -5964, 18576, -10584, -14619, 29484, -18252, -21084, 55188, -28896, -35964, 73008, -49140, -46128, 118692, -54516, -73896, 146340, -95256, -92148, 224694, -111888, -132678, 260064

Offset

0,2

Comments

Cubic AGM theta functions: r(q) (see A004016), s(q) (A005928), t(q) (A005882).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)

Formula

Expansion of (eta(q)^3 / eta(q^3))^4 in powers of q.

Euler transform of period 3 sequence [ -12, -12, -8, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 729 (t/i)^4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A033690.

G.f.: ( Product_{k>0} (1 - x^k)^3 / (1 - x^(3*k)) )^4.

Example

G.f. = 1 - 12*q + 54*q^2 - 84*q^3 - 147*q^4 + 756*q^5 - 756*q^6 - 1212*q^7 + ...

Mathematica

QP = QPochhammer; A133078[n_] := SeriesCoefficient[(QP[q]^3/QP[q^3])^4, {q, 0, n}]; Table[A133078[n], {n, 0, 50}] (* G. C. Greubel, Oct 20 2017 *)

Prog

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

Crossrefs

Cf. A033690.

Sequence in context: A242514 A030182 A060171 * A034436 A186210 A209676

Adjacent sequences: A133075 A133076 A133077 * A133079 A133080 A133081

Keyword

sign

Author

Michael Somos, Sep 08 2007

Status

approved