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%I #20 Mar 12 2021 22:24:46
%S 1,2,-1,0,7,0,-9,0,10,0,-23,0,38,0,-47,0,75,0,-112,0,148,0,-217,0,293,
%T 0,-385,0,553,0,-728,0,928,0,-1272,0,1670,0,-2111,0,2765,0,-3566,0,
%U 4504,0,-5784,0,7300,0,-9123,0,11592,0,-14458,0,17838,0,-22342,0,27668,0,-33884
%N McKay-Thompson series of class 12E 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="/A187197/b187197.txt">Table of n, a(n) for n = -1..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>, Comm. Algebra 22, No. 13, 5175-5193 (1994).
%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 (1/q) * ((psi(q) * psi(q^3)) / (psi(q^2) * psi(q^6)))^2 in powers of q where psi() is a Ramanujan theta function.
%F Expansion of ((eta(q^2)^3 * eta(q^6)^3) / (eta(q) * eta(q^3) * eta(q^4)^2 * eta(q^12)^2))^2 in powers of q.
%F Euler transform of period 12 sequence [ 2, -4, 4, 0, 2, -8, 2, 0, 4, -4, 2, 0, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 4 * g(t) where q = exp(2 Pi i t) and g() is the g.f. for A123861.
%F Given g.f. A(q) then 0 = f(A(q), A(q^2)) where f(u, v) = (v - 4)^2 - u*v * (u - 4). - _Michael Somos_, Aug 31 2014
%F Convolution square of A112165. a(n) = A187196(n) unless n=0. a(2*n) = 0 unless n=0. a(2*n - 1) = A058483(n).
%e G.f. = 1/q + 2 - q + 7*q^3 - 9*q^5 + 10*q^7 - 23*q^9 + 38*q^11 - 47*q^13 + ...
%t a[ n_] := SeriesCoefficient[ (1/q) (QPochhammer[ q^2]^3 QPochhammer[ q^6]^3 / (QPochhammer[ q] QPochhammer[ q^3] QPochhammer[ q^4]^2 QPochhammer[ q^12]^2))^2, {q, 0, n}]; (* _Michael Somos_, Sep 05 2014 *)
%t a[ n_] := SeriesCoefficient[ (1/q) (QPochhammer[ -q] QPochhammer[ -q^3] / (QPochhammer[ q^4] QPochhammer[ q^12]))^2, {q, 0, n}]; (* _Michael Somos_, Sep 05 2014 *)
%t a[ n_] := SeriesCoefficient[ 4 EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^3] / (EllipticTheta[ 2, 0, q] EllipticTheta[ 2, 0, q^3]), {q, 0, n}]; (* _Michael Somos_, Sep 05 2014 *)
%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( ((eta(x^2 + A)^3 * eta(x^6 + A)^3) / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A)^2 * eta(x^12 + A)^2))^2, n))};
%Y Cf. A058483, A112165, A123861, A187196.
%K sign
%O -1,2
%A _Michael Somos_, Mar 06 2011
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