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%I #17 Mar 12 2021 22:24:45
%S 1,-3,1,2,2,-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,
%U -136,-57,-24,-214,-88,201,82,30,308,122,-288,-117,-48,-440,-180,410
%N McKay-Thompson series of class 10E for the Monster group with a(0) = -3.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A139381/b139381.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>
%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F Expansion of eta(q)^3 * eta(q^5) / eta(q^2) / eta(q^10)^3 in powers of q.
%F Expansion of q^(-1) * phi(-q) * f(-q) / (psi(q^5) * f(-q^10)) in powers of q where phi(), psi(), f() are Ramanujan theta functions.
%F Euler transform of period 10 sequence [ -3, -2, -3, -2, -4, -2, -3, -2, -3, 0, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = 20 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A095846.
%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u + 4) * (20 + 6*v) - (v + 4) * (20 + v - u^2).
%F G.f.: (1 / x) * Product_{k>0} (1 - x^k)^3 * (1 - x^(5*k)) / ((1 - x^(2*k)) * (1 - x^(10*k))^3).
%F a(n) = A058101(n) = A132980(n) = A138516(n) unless n=0.
%F Convolution inverse of A095846.
%e G.f. = 1/q - 3 + 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[ -4 + (1/q) QPochhammer[ q^5, q^10]^5 QPochhammer[ -q, q], {q, 0, n}]; (* _Michael Somos_, Sep 07 2015 *)
%t a[ n_] := SeriesCoefficient[ (1/q) QPochhammer[ q]^3 QPochhammer[ q^5] / (QPochhammer[ q^2] QPochhammer[ q^10]^3), {q, 0, n}]; (* _Michael Somos_, Sep 07 2015 *)
%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^5 + A) / eta(x^2 + A) / eta(x^10 + A)^3, n))};
%Y Cf. A058101, A095846, A132980, A138516.
%K sign
%O -1,2
%A _Michael Somos_, Apr 15 2008
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