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

1,2

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

Table of n, a(n) for n=1..35.

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.

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

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 *)

Prog

(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))

Crossrefs

Sequence in context: A260547 A328541 A182818 * A153732 A089924 A293947

Adjacent sequences: A095843 A095844 A095845 * A095847 A095848 A095849

Keyword

nonn

Author

Michael Somos, Jun 08 2004

Status

approved