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McKay-Thompson series of class 34A for Monster.
2

%I #20 Jun 28 2018 12:42:31

%S 1,0,3,2,5,6,12,12,22,22,39,40,63,68,106,112,164,182,257,282,390,432,

%T 584,652,859,964,1253,1404,1794,2024,2556,2880,3594,4054,5016,5662,

%U 6930,7830,9516,10744,12959,14640,17546,19800,23590,26612,31536,35560,41919

%N McKay-Thompson series of class 34A for Monster.

%H G. C. Greubel, <a href="/A058638/b058638.txt">Table of n, a(n) for n = -1..1000</a>

%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 David A. Madore, <a href="http://mathforum.org/kb/thread.jspa?forumID=253&amp;threadID=1602206&amp;messageID=5836094">Coefficients of Moonshine (McKay-Thompson) series</a>, The Math Forum

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

%F Expansion of -1/2 + ( 25/4 + T17A(q) + T17A(q^2) )^(1/2), where T17A(q) = A058530, in powers of q. - _G. C. Greubel_, Jun 24 2018

%F a(n) ~ exp(2*Pi*sqrt(2*n/17)) / (2^(3/4) * 17^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 28 2018

%e T34A = 1/q + 3*q + 2*q^2 + 5*q^3 + 6*q^4 + 12*q^5 + 12*q^6 + 22*q^7 + ...

%t eta[q_] := q^(1/24)*QPochhammer[q]; nmax = 110; A:= q^(1/2)*(eta[q^4]^2 *(eta[q^34]^5/(eta[q]*eta[q^2]*eta[q^17]^3*eta[q^68]^2)) - eta[q^2]^5*(eta[q^68]^2/(eta[q]^3*eta[q^4]^2*eta[q^17]*eta[q^34]))); T17A := (A^2 - 2*q)/q; T34A := -q/2 + q*((25/4) + T17A + (T17A /. {q -> q^2}) + O[q]^nmax)^(1/2); a:= CoefficientList[Series[T34A, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* _G. C. Greubel_, Jun 24 2018 *)

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

%K nonn

%O -1,3

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

%E More terms from _Michel Marcus_, Feb 20 2014