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

Author

Michael Somos, Aug 09 2006

Keywords

• nonn

Comments

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

Examples

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

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.

Crossrefs

Cf. A000726, A030182, A128758.

Programs

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 *)

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))};

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