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McKay-Thompson series of class 22a for Monster.
1

%I #21 Jun 28 2018 03:27:55

%S 1,3,4,11,17,31,45,71,102,158,218,317,440,613,832,1147,1530,2054,2710,

%T 3580,4673,6094,7858,10140,12958,16549,20976,26565,33401,41954,52404,

%U 65365,81119,100534,124048,152820,187578,229848,280708,342262,416056,504943,611202,738590,890379

%N McKay-Thompson series of class 22a for Monster.

%H G. C. Greubel, <a href="/A058569/b058569.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>, 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 (B*(A)^2 + 4*q^3/(B*(A)^2))/(B)^2, where A = q^(1/2)*((eta(q^2)*eta(q^22))^2/(eta(q)*eta(q^4)*eta(q^11)*eta(q^44))) and B = q^(1/2)*(eta(q)*eta(q^11)/(eta(q^2)*eta(q^22))), in powers of q. - _G. C. Greubel_, Jun 21 2018

%F a(n) ~ exp(2*Pi*sqrt(2*n/11)) / (2^(3/4) * 11^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 28 2018

%e T22a = 1/q + 3*q + 4*q^3 + 11*q^5 + 17*q^7 + 31*q^9 + 45*q^11 + 71*q^13 + ...

%t eta[q_]:= q^(1/24)*QPochhammer[q]; e44b := q^(1/2)*(eta[q]*eta[q^11]/(eta[q^2]*eta[q^22])); e88A := q^(1/2)*((eta[q^2]*eta[q^22])^2/(eta[q]*eta[q^4]*eta[q^11]*eta[q^44])); a:= CoefficientList[Series[ (e44b*(e88A)^2 + 4*q^3/(e44b*(e88A)^2))/(e44b)^2, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* _G. C. Greubel_, Jun 21 2018 *)

%o (PARI) q='q+O('q^50); A = ((eta(q^2)*eta(q^22))^2/(eta(q)*eta(q^4)* eta(q^11) *eta(q^44))); B = (eta(q)*eta(q^11)/(eta(q^2)*eta(q^22))); Vec((B*(A)^2 + 4*q^3/(B*(A)^2))/(B)^2) \\ _G. C. Greubel_, Jun 21 2018

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

%K nonn

%O -1,2

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

%E Terms a(12) onward added by _G. C. Greubel_, Jun 21 2018