A121596 Expansion of q^(-1/2)(eta(q^3)/eta(q))^6 in powers of q.

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

Offset: 0

Author

Michael Somos, Aug 09 2006

Keywords

• nonn

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Programs

Mathematica

nmax = 40; CoefficientList[Series[Product[((1-x^(3*k)) / (1-x^k))^6, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 07 2015 *)
CoefficientList[Series[(QPochhammer[q^3]/QPochhammer[q])^6, {q,0,50}], q] (* G. C. Greubel, Nov 02 2018 *)

PARI

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x^3+A)/eta(x+A))^6, n))}

Formula

Euler transform of period 3 sequence [ 6, 6, 0, ...].

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

G.f.: (Product_{k>0} (1-x^(3*k))/(1-x^k))^6.

a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (27 * 2^(3/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015