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A112175 McKay-Thompson series of class 36e for the Monster group.

%I A112175 #19 Jun 26 2025 10:28:11
%S A112175 1,-1,0,-2,2,-1,2,-2,3,-4,4,-4,7,-7,6,-10,11,-11,14,-16,17,-21,22,-24,
%T A112175 32,-34,34,-44,49,-50,60,-66,72,-84,90,-98,117,-125,132,-156,171,-181,
%U A112175 206,-226,245,-277,298,-322,369,-397,422,-480,522,-557,620,-674,728,-807,868,-936,1043,-1121,1198
%N A112175 McKay-Thompson series of class 36e for the Monster group.
%H A112175 G. C. Greubel, <a href="/A112175/b112175.txt">Table of n, a(n) for n = 0..1000</a>
%H A112175 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 A112175 <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F A112175 a(n) = (-1)^n * A112206(n). - _Vaclav Kotesovec_, Jun 06 2018
%F A112175 a(n) ~ (-1)^n * exp(sqrt(2*n)*Pi/3) / (2^(5/4) * sqrt(3) * n^(3/4)). - _Vaclav Kotesovec_, Jun 06 2018
%F A112175 Expansion of q^(1/6)*eta(q)*eta(q^3)/(eta(q^2)*eta(q^6)) in powers of q. - _G. C. Greubel_, Jun 19 2018
%e A112175 T36e = 1/q - q^5 - 2*q^17 + 2*q^23 - q^29 + 2*q^35 - 2*q^41 + 3*q^47 + ...
%t A112175 nmax = 60; CoefficientList[Series[1/Product[(1 + x^(3*k))*(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 06 2018 *)
%t A112175 eta[q_] := q^(1/24)*QPochhammer[q];  a:= CoefficientList[Series[q^(1/6)*(eta[q]*eta[q^3]/(eta[q^2]*eta[q^6])), {q, 0, 60}], q]; Table[a[[n]], {n, 0, 50}] (* _G. C. Greubel_, Jun 19 2018 *)
%o A112175 (PARI) q='q+O('q^60); Vec(eta(q)*eta(q^3)/(eta(q^2)*eta(q^6))) \\ _G. C. Greubel_, Jun 19 2018
%Y A112175 Cf. A112206.
%K A112175 sign
%O A112175 0,4
%A A112175 _Michael Somos_, Aug 28 2005