A121590 Expansion of (eta(q^3) / eta(q))^12 in powers of q.

1, 12, 90, 508, 2391, 9828, 36428, 124188, 395199, 1186344, 3387252, 9257364, 24343037, 61848096, 152356032, 364959196, 852243948, 1944226476, 4341094220, 9502198728, 20419293123, 43131708720, 89656112256, 183580652340

Offset

1,2

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Kevin Acres, David Broadhurst, Eta quotients and Rademacher sums, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.

Formula

Expansion of (c(q) / (3 * b(q)))^3 in powers of q where b(), c() are cubic AGM theta functions. - Michael Somos, Jun 16 2012

Euler transform of period 3 sequence [ 12, 12, 0, ...].

G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^3 + v^3 - u*v - 24*u*v * (u + v) - 729*u^2*v^2.

G.f.: x * (Product_{k>0} (1 - x^(3*k)) / (1 - x^k))^12.

Convolution inverse of A030182. - Michael Somos, Jun 16 2012

G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 3^-6 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A030182.

Convolution 12th power of A000726, cube of A128758, square of A121596. - Michael Somos, Aug 09 2015

a(n) ~ exp(4*Pi*sqrt(n/3)) / (729 * sqrt(2) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015

a(1) = 1, a(n) = (12/(n-1))*Sum_{k=1..n-1} A046913(k)*a(n-k) for n > 1. - Seiichi Manyama, Apr 01 2017

Example

G.f. = q + 12*q^2 + 90*q^3 + 508*q^4 + 2391*q^5 + 9828*q^6 + 36428*q^7 + ...

Mathematica

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^3] / QPochhammer[ q])^12, {q, 0, n}]; (* Michael Somos, Aug 09 2015 *)
nmax = 40; Rest[CoefficientList[Series[x * Product[((1 - x^(3*k)) / (1 - x^k))^12, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Sep 07 2015 *)

Prog

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

Crossrefs

Cf. A000726, A030182, A128758.

Sequence in context: A245577 A156397 A203636 * A341388 A186209 A005758

Adjacent sequences: A121587 A121588 A121589 * A121591 A121592 A121593

Keyword

nonn

Author

Michael Somos, Aug 09 2006

Status

approved