A045490 - OEIS

%I #21 Jun 02 2018 21:27:54

%S 1,4,36,128,386,1024,2488,5632,12031,24576,48308,91904,170110,307200,

%T 542872,941056,1602819,2686976,4439688,7238272,11657090,18561024,

%U 29242240,45617664,70507772,108036096,164192188,247620352,370726652,551215104,814216536,1195226112,1744133125,2530738176

%N McKay-Thompson series of class 8A for Monster.

%H G. C. Greubel, <a href="/A045490/b045490.txt">Table of n, a(n) for n = -1..1000</a>

%H J. H. Conway and S. P. Norton, <a href="http://blms.oxfordjournals.org/content/11/3/308.extract">Monstrous Moonshine</a>, Bull. Lond. Math. Soc. 11 (1979) 308-339.

%H D. Ford, J. McKay and S. P. Norton, <a href="http://dx.doi.org/10.1080/00927879408825127">More on replicable functions</a>, Commun. Algebra 22, No. 13, 5175-5193 (1994).

%H J. McKay and H. Strauss, <a href="http://dx.doi.org/10.1080/00927879008823911">The q-series of monstrous moonshine and the decomposition of the head characters</a>, Comm. Algebra 18 (1990), no. 1, 253-278.

%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>

%F From _Vaclav Kotesovec_, Sep 07 2017: (Start)

%F a(n) = A007265(n) unless n=0.

%F a(n) ~ exp(sqrt(2*n)*Pi) / (2^(5/4) * n^(3/4)). (End)

%F Expansion of -4 + (eta(q^2)*eta(q^4)/(eta(q)*eta(q^8)))^8 in powers of q. - _G. C. Greubel_, Jun 02 2018

%t nmax = 50; CoefficientList[Series[-4*x + Product[((1 - x^(2*k))*(1 - x^(4*k))/((1 - x^k)*(1 - x^(8*k))))^8, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 07 2017 *)

%t eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q*(-4 + (eta[q^2]*eta[q^4]/(eta[q]*eta[q^8]))^8), {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* _G. C. Greubel_, Jun 02 2018 *)

%o (PARI) q='q+O('q^30); a= -4 + (eta(q^2)*eta(q^4)/(eta(q)*eta(q^8)))^8/q; Vec(a) \\ _G. C. Greubel_, Jun 02 2018

%Y Cf. A007265.

%K nonn

%O -1,2

%A _N. J. A. Sloane_