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McKay-Thompson series of class 68A for Monster.
1

%I #18 Jun 29 2018 09:41:02

%S 1,1,1,0,2,1,3,2,4,3,6,4,7,7,10,8,14,12,18,16,23,22,30,28,39,37,49,46,

%T 62,60,78,76,97,96,122,120,150,150,185,184,228,229,278,280,338,342,

%U 410,416,495,506,597,610,718,736,859,884,1026,1058,1224,1262,1453,1505,1722,1784,2039

%N McKay-Thompson series of class 68A for Monster.

%H G. C. Greubel, <a href="/A058742/b058742.txt">Table of n, a(n) for n = -1..2500</a>

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

%F Expansion of sqrt(T34A + 2), where T34A = A058638, in powers of q. - _G. C. Greubel_, Jun 29 2018

%F a(n) ~ exp(2*Pi*sqrt(n/17)) / (2 * 17^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 29 2018

%e T68A = 1/q + q + q^3 + 2*q^7 + q^9 + 3*q^11 + 2*q^13 + 4*q^15 + 3*q^17 + ...

%t QP := QPochhammer; nmax = 260; f[x_, y_] := QP[-x, x*y]*QP[-y, x*y]* QP[x*y, x*y]; G[x_] := f[-x^2, -x^3]/f[-x, -x^2]; H[x_] := f[-x, -x^4]/f[-x, -x^2]; A := G[x^34]*G[x] + x^7*H[x^34]*H[x]; B:= G[x^17]*H[x^2] - x^3*H[x^17]*G[x^2]; T34A := -2 + (A*B)^2/x; a:= CoefficientList[Series[(x*(2 + T34A) + O[x]^nmax)^(1/2), {x, 0, 100}], x]; Table[a[[n]], {n, 1, 80}] (* _G. C. Greubel_, Jun 29 2018 *)

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

%K nonn

%O -1,5

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

%E Terms a(12) onward added by _G. C. Greubel_, Jun 29 2018