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McKay-Thompson series of class 19A for Monster.
5

%I #27 Jun 25 2018 17:39:52

%S 1,0,6,10,21,36,61,96,156,232,357,522,768,1092,1563,2174,3039,4164,

%T 5695,7686,10362,13792,18333,24138,31706,41316,53712,69348,89319,

%U 114396,146114,185724,235482,297252,374316,469578,587646,732888,911961,1131250

%N McKay-Thompson series of class 19A for Monster.

%H G. C. Greubel, <a href="/A058549/b058549.txt">Table of n, a(n) for n = -1..5000</a>

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

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

%F a(n) ~ exp(4*Pi*sqrt(n/19)) / (sqrt(2)*19^(1/4)*n^(3/4)). - _Vaclav Kotesovec_, Sep 07 2017

%F G.f.: - 3 + (1/x) * ( G(x) * G(x^19) + x^4 * H(x) * H(x^19) )^3 where G() is g.f. of A003114 and H() is g.f. of A003106. - _G. C. Greubel_, Jun 25 2018

%e T19A = 1/q + 6*q + 10*q^2 + 21*q^3 + 36*q^4 + 61*q^5 + 96*q^6 + 156*q^7 + ...

%t QP = QPochhammer; G[x_]:= 1/(QP[x, x^5]*QP[x^4, x^5]); H[x_]:= 1/(QP[x^2, x^5]*QP[x^3, x^5]); a:= CoefficientList[Series[-3 *x + (G[x]*G[x^19] + x^4*H[x]*H[x^19])^3, {x, 0, 60}], x]; Table[a[[n]], {n, 1, 50}] (* _G. C. Greubel_, Jun 14 2018 *)

%o (PARI) q='q+O('q^40); A = (eta(q)*eta(q^19)/(eta(q^2)*eta(q^38)))^2; B = - (eta(-q)*eta(-q^19)/(eta(q^2)*eta(q^38)))^2; F = (q^4/(A*B) - (A + B)/(4*q))^3; v=Vec(F-3*q^2); vector(#v\2, n, v[2*n-1]) \\ _G. C. Greubel_, Jun 25 2018

%Y Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.

%Y Cf. A136569 (same sequence except for n=0).

%K nonn

%O -1,3

%A _N. J. A. Sloane_, Nov 27 2000

%E More terms from _Michel Marcus_, Feb 18 2014