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%I #35 Jun 14 2018 20:24:55
%S 1,-1,-1,-1,2,1,-2,1,3,0,-4,-1,5,-1,-7,0,8,0,-10,1,13,2,-16,0,20,-3,
%T -24,-2,30,2,-36,4,43,0,-52,-3,61,-2,-73,-1,86,1,-102,3,120,4,-140,-1,
%U 165,-8,-192,-5,224,6,-260,10,301,2,-348,-7,401,-7,-462,-2,530,2,-608
%N McKay-Thompson series of class 24D for the Monster group.
%C Ramanujan theta functions: f(q) := Product_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Product_{k>=0} (1+q^(2k+1)) (A000700).
%H G. C. Greubel, <a href="/A058574/b058574.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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F Expansion of chi(-x) * chi(-x^2) * chi(-x^3) * chi(-x^6) in powers of x where chi() is a Ramanujan theta function. - _Michael Somos_, Mar 06 2011
%F Expansion of q^(1/2) * eta(q) * eta(q^3) / (eta(q^4) * eta(q^12)) in powers of q. - _Michael Somos_, Mar 06 2011
%F Euler transform of period 12 sequence [ -1, -1, -2, 0, -1, -2, -1, 0, -2, -1, -1, 0, ...]. - _Michael Somos_, Mar 06 2011
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 4 / f(t) where q = exp(2 Pi i t). - _Michael Somos_, Mar 06 2011
%F Convolution square is A187196. a(n) = (-1)^n * A112165(n). - _Michael Somos_, Mar 06 2011
%e T24D = 1/q - q - q^3 - q^5 + 2*q^7 + q^9 - 2*q^11 + q^13 + 3*q^15 - 4*q^19 + ...
%t QP = QPochhammer; s = QP[q]*(QP[q^3]/(QP[q^4]*QP[q^12])) + O[q]^70; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 13 2015, from 2nd formula *)
%t eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q^(1/2)* eta[q]*eta[q^3]/(eta[q^4]*eta[q^12]), {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* _G. C. Greubel_, Jun 14 2018 *)
%o (PARI) q='q+O('q^50); A=eta(q)*eta(q^3)/(eta(q^4)*eta(q^12)); Vec(A) \\ _G. C. Greubel_, Jun 14 2018
%Y Cf. A000521, A007240, A007241, A007267, A014708, A045478.
%Y Cf. A112165, A187196. - _Michael Somos_, Mar 06 2011
%K sign
%O 0,5
%A _N. J. A. Sloane_, Nov 27 2000
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