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A112142 McKay-Thompson series of class 8B for the Monster group.

%I A112142 #37 Feb 16 2025 08:32:58
%S A112142 1,12,66,232,639,1596,3774,8328,17283,34520,66882,125568,229244,
%T A112142 409236,716412,1231048,2079237,3459264,5677832,9200232,14729592,
%U A112142 23325752,36567222,56778888,87369483,133315692,201825420,303257512
%N A112142 McKay-Thompson series of class 8B for the Monster group.
%C A112142 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H A112142 Seiichi Manyama, <a href="/A112142/b112142.txt">Table of n, a(n) for n = 0..1000</a>
%H A112142 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 A112142 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A112142 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A112142 <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F A112142 Expansion of chi(q)^12 in powers of q where chi() is a Ramanujan theta function.
%F A112142 Expansion of q^(1/2) * (eta(q^2)^2 / (eta(q) * eta(q^4)))^12 in powers of q.
%F A112142 G.f.: Product_{k>0} (1 + (-x)^k)^-12 = Product_{k>0} (1 + x^(2*k - 1))^-12.
%F A112142 a(n) = (-1)^n * A007249(n). Convolution inverse of A124863.
%F A112142 G.f.: T(0), where T(k) = 1 - 1/(1 - 1/(1 - 1/(1+(x)^(2*k+1))^12/T(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Nov 06 2013
%F A112142 a(n) ~ exp(Pi*sqrt(2*n)) / (2^(5/4) * n^(3/4)). - _Vaclav Kotesovec_, Aug 27 2015
%F A112142 G.f.: exp(12*Sum_{k>=1} x^k/(k*(1 - (-x)^k))). - _Ilya Gutkovskiy_, Jun 07 2018
%e A112142 1 + 12*x + 66*x^2 + 232*x^3 + 639*x^4 + 1596*x^5 + 3774*x^6 + 8328*x^7 + ...
%e A112142 T8B = 1/q + 12*q + 66*q^3 + 232*q^5 + 639*q^7 + 1596*q^9 + 3774*q^11 + ...
%t A112142 a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ ((1 - m) m / 16 / q)^(1/2), {q, 0, n}]] (* _Michael Somos_, Jul 22 2011 *)
%t A112142 a[ n_] := SeriesCoefficient[ Product[ 1 + x^k, {k, 1, n, 2}]^-12, {x, 0, n}] (* _Michael Somos_, Jul 22 2011 *)
%t A112142 nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k+1))^12, {k, 0, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Aug 27 2015 *)
%t A112142 QP = QPochhammer; s = (QP[q^2]^2/(QP[q]*QP[q^4]))^12 + O[q]^50; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 16 2015, adapted from PARI *)
%o A112142 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x^2 + A)^2 / (eta(x + A) * eta(x^4 + A)))^12, n))}
%Y A112142 Cf. A007249, A124863.
%K A112142 nonn
%O A112142 0,2
%A A112142 _Michael Somos_, Aug 28 2005