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A058487 McKay-Thompson series of class 12I for the Monster group.

%I A058487 #43 Feb 16 2025 08:32:43
%S A058487 1,2,1,0,-2,-2,2,4,3,-4,-8,-4,5,14,7,-8,-20,-12,14,28,17,-20,-44,-24,
%T A058487 28,66,36,-40,-90,-52,56,124,71,-80,-176,-96,109,244,133,-144,-326,
%U A058487 -182,198,432,240,-268,-580,-316,349,772,420,-456,-1004,-552,600,1300,713,-780,-1692,-916,1001,2186,1182
%N A058487 McKay-Thompson series of class 12I for the Monster group.
%C A058487 Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C A058487 Number 5 of the 15 generalized eta-quotients listed in Table I of Yang 2004. - _Michael Somos_, Jul 21 2014
%C A058487 A generator (Hauptmodul) of the function field associated with congruence subgroup Gamma_0(12). [Yang 2004] - _Michael Somos_, Jul 21 2014
%H A058487 G. C. Greubel, <a href="/A058487/b058487.txt">Table of n, a(n) for n = 0..5000</a> (terms 0..502 from G. A. Edgar)
%H A058487 D. Ford, J. McKay and S. P. Norton, <a href="http://dx.doi.org/10.1080/00927879408825127">More on replicable functions</a>, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%H A058487 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 A058487 Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>
%H A058487 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>
%H A058487 Y. Yang, <a href="http://dx.doi.org/10.1112/S0024609304003510">Transformation formulas for generalized Dedekind eta functions</a>, Bull. London Math. Soc. 36 (2004), no. 5, 671-682. See p. 679, Table 1.
%H A058487 <a href="/index/Gre#groups">Index entries for sequences related to groups</a>
%H A058487 <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F A058487 Expansion of (psi(x) / psi(x^3))^2 in powers of x where psi() is a Ramanujan theta function. - _Michael Somos_, Jul 21 2014
%F A058487 G.f.: ( Product_{k>0} (1 - x^(6*k - 2)) * (1 - x^(6*k - 4)) / ((1 - x^(6*k - 1)) * (1 - x^(6*k - 5))) )^2.
%F A058487 Expansion of q^(1/2) * (eta(q^2)^4 * eta(q^3)^2 / (eta(q)^2 * eta(q^6)^4)) in powers of q.
%F A058487 Euler transform of period 6 sequence [ 2, -2, 0, -2, 2, 0,...]. - _Michael Somos_, Mar 18 2004
%F A058487 Given g.f. A(x), then B(q) = A(q^2) / q satisfies 0 = f(B(x), B(x^2)) where f(u, v) = u^2 + 3*v - u^2*v + v^2. - _Michael Somos_, Mar 18 2004
%F A058487 G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 3 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A186924.
%F A058487 a(n) = (-1)^n * A062243(n).
%F A058487 Convolution square is A128633. Convolution inverse of A217786. - _Michael Somos_, Jul 21 2014
%e A058487 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 A058487 T12I = 1/q + 2*q + 1*q^3 - 2*q^7 - 2*q^9 + 2*q^11 + 4*q^13 + 3*q^15 - 4*q^17 + ...
%t A058487 a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, q]^2 / EllipticTheta[ 2, 0, q^3]^2, {q, 0, 2 n - 1}]; (* _Michael Somos_, Jul 21 2014 *)
%t A058487 QP = QPochhammer; s = QP[q^2]^4*(QP[q^3]^2/(QP[q]^2*QP[q^6]^4)) + O[q]^70; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 12 2015 *)
%o A058487 (PARI) {a(n) = local(A, m); if( n<0, 0, A = 1 + O(x); m=1; while( m<=n, m*=2; A = subst(A, x, x^2); A = sqrt(A * (A + 3*x) / (A - x))); polcoeff(A, n))};
%o A058487 (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 * eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)^2))^2, n))};
%Y A058487 Cf. A000521, A007240, A014708, A007241, A007267, A045478, A062243, A128633, A186924, A217786.
%K A058487 sign
%O A058487 0,2
%A A058487 _N. J. A. Sloane_, Nov 27 2000