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%I #19 Mar 12 2021 22:24:46
%S 1,-2,0,0,2,0,0,0,-1,0,0,0,-2,0,0,0,3,0,0,0,2,0,0,0,-4,0,0,0,-4,0,0,0,
%T 5,0,0,0,8,0,0,0,-8,0,0,0,-10,0,0,0,11,0,0,0,12,0,0,0,-15,0,0,0,-18,0,
%U 0,0,22,0,0,0,26,0,0,0,-29,0,0,0,-34,0,0,0
%N McKay-Thompson series of class 16B 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).
%C Number 6 of the 15 generalized eta-quotients listed in Table I of Yang 2004. - _Michael Somos_, Jul 21 2014
%C A generator (Hauptmodul) of the function field associated with congruence subgroup Gamma_0(16). [Yang 2004] - _Michael Somos_, Jul 21 2014
%H G. C. Greubel, <a href="/A185338/b185338.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 W. Weisstein, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">MathWorld: Ramanujan Theta Functions</a>
%H 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.
%F Expansion of q^(-1) * phi(-q) / psi(q^8) in powers of q where phi(), psi() are Ramanujan theta functions.
%F Expansion of eta(q)^2 * eta(q^8) / (eta(q^2) * eta(q^16)^2) in powers of q.
%F Euler transform of period 16 sequence [ -2, -1, -2, -1, -2, -1, -2, -2, -2, -1, -2, -1, -2, -1, -2, 0, ...].
%F G.f. A(q) satisfies 0 = f(A(q), A(q^2)) where f(u, v) = v^2 - u * (u + 4) * (v + 2).
%F G.f. A(q) satisfies 0 = f(A(q), A(q^3)) where f(u, v) = (u - v)^4 - u * (u^2 + 6*u + 8) * v * (v^2 + 6*v + 8).
%F G.f. A(q) satisfies 0 = f(A(q), A(q^4)) where f(u, v) = v^4 - u * (u + 4) * (u^2 + 4*u + 8) * (v + 2) * (v^2 + 4*v + 8).
%F G.f. A(q) satisfies 0 = f(A(q), A(q^5)) where f(u, v) = (u - v)^6 - u * (u + 2) * (u + 4) * (u^2 + 4*u + 8) * v * (v + 2) * (v + 4) * (v^2 + 4*v + 8).
%F G.f. A(q) satisfies 0 = f(A(q), A(q^2), A(q^3), A(q^6)) where f(u1, u2, u3, u6) = 2 * (u1 - u3)^2 - 2 * u2*u6 + u2*u3 * (u3 + 4) + u1*u6 * (u1 + 4).
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 8 g(t) where q = exp(2 Pi i t) and g() is the g.f. of A123655.
%F G.f.: -2 + (1/q) * Product_{k>0} ((1 + q^(8*k - 4)) / (1 + q^(8*k)))^2.
%F a(4*n - 1) = A029839(n). a(4*n) = 0 unless n=0. a(4*n + 1) = a(4*n + 2) = 0.
%F Convolution inverse of A123655.
%e G.f. = 1/q - 2 + 2*q^3 - q^7 - 2*q^11 + 3*q^15 + 2*q^19 - 4*q^23 - 4*q^27 + 5*q^31 + ...
%t a[ n_] := SeriesCoefficient[ 2 EllipticTheta[ 4, 0, q] / EllipticTheta[ 2, 0, q^4], {q, 0, n}]; (* _Michael Somos_, Jul 21 2014 *)
%o (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^8 + A) / (eta(x^2 + A) * eta(x^16 + A)^2), n))};
%Y Cf. A029839, A123655.
%K sign
%O -1,2
%A _Michael Somos_, Feb 28 2012
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