A093065 Expansion of (eta(q^3)eta(q^15)/(eta(q)eta(q^5)))^2 in powers of q.

0, 1, 2, 5, 8, 16, 28, 48, 78, 124, 194, 302, 454, 682, 996, 1457, 2096, 2993, 4226, 5920, 8228, 11373, 15580, 21246, 28740, 38731, 51872, 69155, 91716, 121105, 159208, 208512, 271894, 353338, 457336, 590124, 758792, 972677, 1242896, 1583576

Offset

0,3

Comments

Euler transform of period 15 sequence [2,2,0,2,4,0,2,2,0,4,2,0,2,2,0,...].

Links

Table of n, a(n) for n=0..39.

Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015.

Formula

G.f. A(x) satisfies 0=f(A(x), A(x^2)) where f(u, v)=u^3+v^3-4uv(u+v)-9u^2v^2-uv.

a(n) ~ exp(4*Pi*sqrt(n/15)) / (9*sqrt(2)*15^(1/4)*n^(3/4)). - Vaclav Kotesovec, Jul 11 2016

Mathematica

nmax = 50; CoefficientList[Series[x*Product[((1-x^(3*k)) * (1-x^(15*k)) / ((1-x^k) * (1-x^(5*k))))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 11 2016 *)

Prog

(PARI) {a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff((eta(x^3+A)*eta(x^15+A)/eta(x+A)/eta(x^5+A))^2, n))}
(PARI) {a(n)=local(A, u, v); if(n<0, 0, A=x; for(k=2, n, u=A+x*O(x^k); v=subst(u, x, x^2); A-=x^k*polcoeff(u^3+v^3-4*u*v*(u+v)-9*u^2*v^2-u*v, k+2)/2); polcoeff(A, n))}

Crossrefs

Sequence in context: A328547 A137685 A169826 * A301596 A026007 A032233

Adjacent sequences: A093062 A093063 A093064 * A093066 A093067 A093068

Keyword

nonn

Author

Michael Somos, Mar 17 2004

Extensions

Changed offset to 0 and added a(0)=0 by Vaclav Kotesovec, Jul 11 2016

Status

approved