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%I #28 Mar 12 2021 22:24:42
%S 1,-2,1,0,-2,2,2,-4,3,4,-8,4,5,-14,7,8,-20,12,14,-28,17,20,-44,24,28,
%T -66,36,40,-90,52,56,-124,71,80,-176,96,109,-244,133,144,-326,182,198,
%U -432,240,268,-580,316,349,-772,420,456,-1004,552,600,-1300,713,780,-1692,916,1001,-2186,1182
%N McKay-Thompson series of class 24c for the Monster group.
%C Expansion of a Hauptmodul for Gamma'_0(12).
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A062243/b062243.txt">Table of n, a(n) for n = 0..1000</a>
%H J. McKay and A. Sebbar, <a href="http://dx.doi.org/10.1007/s002080000116">Fuchsian groups, automorphic functions and Schwarzians</a>, Math. Ann., 318 (2000), 255-275.
%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 Euler transform of period 12 sequence [-2, 0, 0, -2, -2, 0, -2, -2, 0, 0, -2, 0, ...]. - _Michael Somos_, May 14 2004
%F G.f.: ( Product_{k>0} (1 - x^(4*k)) * (1 - x^(2*k-1)) / (1 - x^(3*k)) )^2.
%F Given G.f. A(x), then B(q) = A(q^2)^2 / (3*q^2) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (u + v)^2 * (u^2 + v^2 - u*v) + 3 * (u^3 + v^3) * (1 + u*v) - 9*u*v * (1 + (u*v)^2) - 90*(u*v)^2 - 27*u*v * (u + v) * (1 + u*v). - _Michael Somos_, May 14 2004
%F Expansion of q^(1/2) * (eta(q) * eta(q^4) * eta(q^6) / (eta(q^2) * eta(q^3) * eta(q^12)))^2 in powers of q.
%F a(n) = (-1)^n * A058487(n).
%F Power series expansion of f(-x^2)^2 / f(x, x^5)^2 = psi(-x)^2 / psi(-x^3)^2 in powers of x where f(), psi() are Ramanujan theta functions and f(, ) is Ramanujan's general theta function. - _Michael Somos_, Oct 22 2017
%e G.f. = 1 - 2*x + x^2 - 2*x^4 + 2*x^5 + 2*x^6 - 4*x^7 + 3*x^8 + 4*x^9 - 8*x^10 + ...
%e T24c = 1/q - 2*q + q^3 - 2*q^7 + 2*q^9 + 2*q^11 - 4*q^13 + 3*q^15 + 4*x^17 + ...
%t a[ n_] := SeriesCoefficient[ x^(1/2) EllipticTheta[ 2, Pi/4, x^(1/2)]^2 / EllipticTheta[ 2, Pi/4, x^(3/2)]^2, {x, 0, n}]; (* _Michael Somos_, Oct 22 2017 *)
%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) * eta(x^6 + A) / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)))^2, n))};
%Y Cf. A058487.
%K sign
%O 0,2
%A _N. J. A. Sloane_, Jul 01 2001
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