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%I #16 Mar 06 2020 20:32:41
%S 1,-1,-2,1,0,2,1,0,0,-1,0,-4,-1,0,4,0,0,2,1,0,-8,2,0,8,0,0,2,-2,0,-16,
%T -3,0,16,-1,0,4,4,0,-28,4,0,28,1,0,8,-4,0,-48,-6,0,46,-1,0,12,5,0,-80,
%U 8,0,76,1,0,20,-8,0,-126,-10,0,120,-2,0,32,11,0,-196,14,0,184,4,0,48
%N McKay-Thompson series of class 18A for the Monster group with a(0) = -1.
%H G. C. Greubel, <a href="/A093073/b093073.txt">Table of n, a(n) for n = -1..1000</a>
%F Expansion of eta(q) * eta(q^2) / (eta(q^9) * eta(q^18)) in powers of q.
%F Euler transform of period 18 sequence [ -1, -2, -1, -2, -1, -2, -1, -2, 0, -2, -1, -2, -1, -2, -1, -2, -1, 0, ...]. - Corrected by _Sean A. Irvine_, Mar 06 2020
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = u^4 + v^4 - u*v * ((u + v)^2 + 9*(u + v) + u*v * (u + v + 4)).
%F a(3*n - 1) = A062242(n), a(3*n + 1) = -2*A092848(n). a(3*n) = 0, unless n=0. a(n) = A058531(n) unless n=0.
%e G.f. = 1/q - 1 - 2*q + q^2 + 2*q^4 + q^5 - q^8 - 4*q^10 - q^11 + 4*q^13 + ...
%t a[ n_] := SeriesCoefficient[ q^(-1) QPochhammer[ q] QPochhammer[ q^2] / (QPochhammer[ q^9] QPochhammer[ q^18]), {q, 0, n}] (* _Michael Somos_, Oct 23 2013 *)
%o (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x + x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A) / (eta(x^9 + A) * eta(x^18 + A)), n))}
%Y Cf. A062242, A092848, A058531.
%K sign
%O -1,3
%A _Michael Somos_, Mar 17 2004
%E Edited for better readability and coherence. - _Michael Somos_, Oct 23 2013
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