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%I #20 Mar 12 2021 22:24:46
%S 1,-1,1,-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
%N McKay-Thompson series of class 20C for the Monster group with a(0) = -1.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A223903/b223903.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 q^(-1) * chi(q^5)^5 / chi(q) in powers of q where chi() is a Ramanujan theta function.
%F Expansion of eta(q) * eta(q^4) * eta(q^10)^10 / (eta(q^2)^2 * eta(q^5)^5 * eta(q^20)^5) in powers of q.
%F Euler transform of period 20 sequence [ -1, 1, -1, 0, 4, 1, -1, 0, -1, -4, -1, 0, -1, 1, 4, 0, -1, 1, -1, 0, ...].
%F G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v)^4 - u * (u - 1) * (u + 4) * v * (v - 1) * (v + 4).
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. of A225701. - _Michael Somos_, Sep 04 2013
%F G.f.: (1/x) * Product_{k>0} (1 + x^(10*k - 5))^5 / (1 + x^(2*k - 1)).
%F a(n) = A112159(n) = A145740(n) unless n=0. a(n) = -(-1)^n * A132980(n).
%e G.f. = 1/q - 1 + 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[ (1/q) QPochhammer[ -q^5, q^10]^5 / QPochhammer[ -q, q^2], {q, 0, n}];
%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)^10 / (eta(x^2 + A)^2 * eta(x^5 + A)^5 * eta(x^20 + A)^5), n))};
%Y Cf. A112159, A132980, A145740, A225701.
%K sign
%O -1,4
%A _Michael Somos_, Mar 29 2013
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