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%I #23 Mar 12 2021 22:24:45
%S 1,-2,1,-2,2,2,-1,0,-4,2,5,-2,0,-8,2,8,-3,2,-14,6,14,-6,4,-24,12,24,
%T -11,4,-40,16,38,-16,5,-62,24,60,-24,10,-94,40,91,-38,18,-144,62,136,
%U -57,24,-214,88,201,-82,30,-308,122,288,-117,48,-440,180,410,-168
%N McKay-Thompson series of class 20C for the Monster group with a(0) = -2.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A145740/b145740.txt">Table of n, a(n) for n = -1..1000</a>
%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%F Expansion of (eta(q) * eta(q^4) * eta(q^10) / (eta(q^2) * eta(q^5) * eta(q^20)))^2 in powers of q.
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (80 t)) = 5 / f(t) where q = exp(2 Pi i t).
%F Expansion of q^(-1) * (psi(-q) / psi(-q^5))^2 in powers of q where psi() is a Ramanujan theta function.
%F a(n) = A112159(n) = A225849(n) unless n=0.
%F a(2*n) = -2 * A138522(n).
%F Convolution square of A145708.
%F a(n) = -(-1)^n * A138516(n). - _Michael Somos_, Sep 04 2015
%e G.f. = 1/q - 2 + q - 2*q^2 + 2*q^3 + 2*q^4 - q^5 - 4*q^7 + 2*q^8 + 5*q^9 + ...
%t a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, Pi/4, q^(1/2)] / EllipticTheta[ 2, Pi/4, q^(5/2)])^2, {q, 0, n}]; (* _Michael Somos_, Sep 04 2015 *)
%t QP = QPochhammer; s = (QP[q]*QP[q^4]*(QP[q^10]/(QP[q^2]*QP[q^5]*QP[q^20]) ))^2 + O[q]^60; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 15 2015, adapted from PARI *)
%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) * eta(x^10 + A) / (eta(x^2 + A) * eta(x^5 + A) * eta(x^20 + A)))^2, n))};
%Y Cf. A112159, A138561, A138522, A145708, A225849.
%K sign
%O -1,2
%A _Michael Somos_, Oct 17 2008
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