A186209 Coefficients of modular function denoted g_5(tau) by Atkin.
1, 12, 90, 520, 2535, 10908, 42614, 153960, 521235, 1669720, 5098938, 14931060, 42124236, 114944700, 304344780, 784057428, 1969912725, 4836549432, 11624458120, 27391979940, 63368714904, 144094899520, 322411353540
Offset: 5
Keywords
Examples
G.f. = q^5 + 12*q^6 + 90*q^7 + 520*q^8 + 2535*q^9 + 10908*q^10 + 42614*q^11 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 5..2500
- A. O. L. Atkin, Proof of a conjecture of Ramanujan, Glasgow Math. J., 8 (1967), 14-32.
- 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
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Mathematica
nmax=40; Drop[CoefficientList[Series[x^5*Product[((1-x^(11*k)) / (1-x^k))^12,{k,1,nmax}],{x,0,nmax}],x], 5] (* Vaclav Kotesovec, Oct 14 2015 *) eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[(eta[q^11] /eta[q])^12, {q, 0, 50}], q] (* G. C. Greubel, Aug 14 2018 *) -
PARI
{a(n) = my(A); if( n<5, 0, n-=5; A = x * O(x^n); polcoeff( (eta(x^11 + A) / eta(x + A))^12, n))};
Formula
Expansion of (eta(q^11) / eta(q))^12 in powers of q.
Euler transform of period 11 sequence [12, 12, 12, 12, 12, 12, 12, 12, 12, 12, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (11 t)) = 11^-6 / f(t) where q = exp(2 Pi i t).
G.f.: x^5 * (Product_{k>0} (1 - x^(11*k)) / (1 - x^k))^12.
a(n) ~ 5^(1/4) * exp(4*Pi*sqrt(5*n/11)) / (sqrt(2) * 11^(25/4) * n^(3/4)). - Vaclav Kotesovec, Oct 14 2015