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%I #15 Mar 12 2021 22:24:46
%S 1,5,6,-4,-3,12,-8,-12,30,-20,-30,72,-46,-60,156,-96,-117,300,-188,
%T -228,552,-344,-420,1008,-603,-732,1770,-1048,-1245,2976,-1776,-2088,
%U 4908,-2900,-3420,7992,-4658,-5460,12756,-7408,-8583,19944,-11564,-13344,30756
%N McKay-Thompson series of class 12B for the Monster group with a(0) = 5.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. A. Edgar, <a href="/A187146/b187146.txt">Table of n, a(n) for n = -1..1002</a>
%H 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 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) * (phi(q)^3 * psi(-q)) / (phi(q^3) * psi(-q^3)^3) in powers of q where phi(), psi() are Ramanujan theta functions.
%F Expansion of eta(q^2)^14 / (eta(q)^5 * eta(q^3) * eta(q^4)^5 * eta(q^6)^2 * eta(q^12)) in powers of q.
%F Euler transform of period 12 sequence [ 5, -9, 6, -4, 5, -6, 5, -4, 6, -9, 5, 0, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 9 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A164617.
%F Convolution of A113660 and A133637.
%F a(n) = -(-1)^n * A128632(n). - _Michael Somos_, May 20 2015
%e G.f. = 1/q + 5 + 6*q - 4*q^2 - 3*q^3 + 12*q^4 - 8*q^5 - 12*q^6 + 30*q^7 - 20*q^8 + ...
%t a[ n_] := SeriesCoefficient[ 2 EllipticTheta[ 3, 0, q]^3 EllipticTheta[ 2, Pi/4, q^(1/2)] / (EllipticTheta[ 3, 0, q^3] EllipticTheta[ 2, Pi/4, q^(3/2)]^3), {q, 0, n}]; (* _Michael Somos_, May 20 2015 *)
%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^2 + A)^14 / (eta(x + A)^5 * eta(x^3 + A) * eta(x^4 + A)^5 * eta(x^6 + A)^2 * eta(x^12 + A)), n))};
%Y Cf. A112148, A113660, A128632, A133637, A164617.
%K sign
%O -1,2
%A _Michael Somos_, Mar 05 2011
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