A095846 - OEIS

%I #10 Oct 18 2018 03:05:48

%S 1,3,8,19,41,84,164,307,557,983,1692,2852,4718,7672,12288,19411,30274,

%T 46671,71180,107479,160792,238476,350828,512196,742441,1068914,

%U 1529120,2174216,3073670,4321444,6044072,8411283,11649936,16062102,22048604

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

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

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

%H Kevin Acres, David Broadhurst, <a href="https://arxiv.org/abs/1810.07478">Eta quotients and Rademacher sums</a>, arXiv:1810.07478 [math.NT], 2018. See Table 1 p. 10.

%H Vaclav Kotesovec, <a href="http://arxiv.org/abs/1509.08708">A method of finding the asymptotics of q-series based on the convolution of generating functions</a>, arXiv:1509.08708 [math.CO], 2015-2016.

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

%F a(n) ~ exp(2*Pi*sqrt(2*n/5)) / (2^(11/4) * 5^(5/4) * n^(3/4)). - _Vaclav Kotesovec_, Oct 13 2015

%t 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 *)

%o (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))

%K nonn

%O 1,2

%A _Michael Somos_, Jun 08 2004