A045489 - OEIS

%I #30 May 29 2018 16:44:41

%S 1,3,51,204,681,1956,5135,12360,28119,60572,125682,251040,487426,

%T 920568,1699611,3070508,5445510,9490116,16283793,27537708,45959775,

%U 75760640,123471327,199081632,317814988

%N McKay-Thompson series of class 7A for the Monster group with a(0) = 3.

%H G. C. Greubel, <a href="/A045489/b045489.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 N. D. Elkies, <a href="http://www.math.harvard.edu/~elkies/modular.pdf">Elliptic and modular curves over finite fields and related computational issues</a>, in AMS/IP Studies in Advanced Math., 7 (1998), 21-76, esp. p. 66.

%H D. Ford, J. McKay and S. P. Norton, <a href="http://dx.doi.org/10.1080/00927879408825127">More on replicable functions</a>, Comm. 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 Expansion of -7 + (h+7)^2/h, where h = (eta(q)/eta(q^7))^4.

%F a(n) ~ exp(4*Pi*sqrt(n/7)) / (sqrt(2) * 7^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2017

%e 1/q + 3 + 51*q + 204*q^2 + 681*q^3 + 1956*q^4 + 5135*q^5 + 12360*q^6 + ...

%t QP = QPochhammer; h = q*(QP[q^7]/QP[q])^4; s = 1 - 7*q + q*((1+7*h)^2/h - 1/q) + O[q]^30; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 15 2015 *)

%t eta[q_]:= q^(1/24)*QPochhammer[q]; h:= (eta[q]/eta[q^7])^4; A045489 := CoefficientList[Series[q*(h + 7 + 49/h), {q, 0, 50}], q]; Table[ A045489[[n]], {n, 1, 30}] (* _G. C. Greubel_, May 28 2018 *)

%o (PARI) q='q+O('q^30); {h =(eta(q)/eta(q^7))^4/q};  Vec(h + 7 + 49/h) \\ _G. C. Greubel_, May 28 2018

%Y Essentially same as A007264 and A030183.

%K nonn

%O -1,2

%A _N. J. A. Sloane_