A095846 Expansion of eta(q^2)eta(q^10)^3/(eta(q^5)eta(q)^3) in powers of q.

1, 3, 8, 19, 41, 84, 164, 307, 557, 983, 1692, 2852, 4718, 7672, 12288, 19411, 30274, 46671, 71180, 107479, 160792, 238476, 350828, 512196, 742441, 1068914, 1529120, 2174216, 3073670, 4321444, 6044072, 8411283, 11649936, 16062102, 22048604

Offset: 1

Author

Michael Somos, Jun 08 2004

Keywords

• nonn

Comments

G.f. A(x) satisfies 0=f(A(x),A(x^2)) where f(u,v)=-u^2+v+6uv+4v^2+20uv^2.

Euler transform of period 10 sequence [3,2,3,2,4,2,3,2,3,0,...].

Links

• Kevin Acres, David Broadhurst, Eta quotients and Rademacher sums, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.
• Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.

Programs

Mathematica

nmax=60; Rest[CoefficientList[Series[x*Product[(1+x^k) * (1-x^(5*k))^2 * (1+x^(5*k))^3 / (1-x^k)^2,{k,1,nmax}],{x,0,nmax}],x]] (* Vaclav Kotesovec, Oct 13 2015 *)

PARI

a(n)=local(A); if(n<1,0,n--; A=x*O(x^n); polcoeff(eta(x^2+A)*eta(x^10+A)^3/(eta(x+A)^3*eta(x^5+A)),n))

Formula

G.f.: x*(Product_{k>0} (1-x^(2k))(1-x^(10k))^3/((1-x^k)^3(1-x^(5k)))).

a(n) ~ exp(2*Pi*sqrt(2*n/5)) / (2^(11/4) * 5^(5/4) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015