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McKay-Thompson series of class 72a for the Monster group.
2

%I #19 Jun 16 2018 18:32:50

%S 1,1,1,0,1,2,2,2,3,4,4,4,5,7,7,8,10,12,14,14,17,20,22,24,28,33,36,40,

%T 45,52,56,62,71,80,88,96,109,122,133,144,163,182,198,216,240,268,290,

%U 316,349,386,420,456,502,552,600,650,713,780,846,916,1001,1093,1182

%N McKay-Thompson series of class 72a for the Monster group.

%H G. C. Greubel, <a href="/A112205/b112205.txt">Table of n, a(n) for n = 0..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 <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>

%F a(n) ~ exp(Pi*sqrt(2*n)/3) / (2^(5/4)*sqrt(3)*n^(3/4)). - _Vaclav Kotesovec_, Nov 12 2015

%F Expansion of q^(1/2)*(eta(q^2)*eta(q^3)^2*eta(q^12)^2*eta(q^18)/(eta(q) *eta(q^4)*eta(q^6)^2*eta(q^9)*eta(q^36))) in powers of q. - _G. C. Greubel_, Jun 16 2018

%e T72a = 1/q + q + q^3 + q^7 + 2*q^9 + 2*q^11 + 2*q^13 + 3*q^15 + 4*q^17 + ...

%t nmax = 60; CoefficientList[Series[Product[(1+x^k) * (1-x^(3*k))^2 * (1+x^(6*k))^2 * (1+x^(9*k)) / ((1-x^(4*k)) * (1-x^(36*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Nov 12 2015 *)

%t eta[q_]:= q^(1/24)*QPochhammer[q]; e72a:= q^(1/2)*((eta[q^2]*eta[q^3]^2 *eta[q^12]^2*eta[q^18])/(eta[q]*eta[q^4]*eta[q^6]^2*eta[q^9]*eta[q^36])); a[n_]:= SeriesCoefficient[e72a, {q, 0, n}]; Table[a[n], {n,0,50}] (* _G. C. Greubel_, Feb 18 2018 *)

%o (PARI) q='q+O('q^50); A = (eta(q^2)*eta(q^3)^2*eta(q^12)^2*eta(q^18)/( eta(q)*eta(q^4)*eta(q^6)^2*eta(q^9)*eta(q^36))); Vec(A) \\ _G. C. Greubel_, Jun 16 2018

%K nonn

%O 0,6

%A _Michael Somos_, Aug 28 2005