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%I #23 Mar 12 2021 22:24:43
%S 1,0,6,-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,-17744,-20448,46944
%N McKay-Thompson series of class 12B for the Monster group.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%H G. C. Greubel, <a href="/A112148/b112148.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>, 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>
%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F Expansion of -5 + (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 a(n) = -(-1)^n * A007258(n). - _Michael Somos_, May 20 2015
%F a(n) = A187146(n) = A187147(n) = A187148(n) unless n=0. - _Michael Somos_, May 20 2015
%e T12B = 1/q + 6*q - 4*q^2 - 3*q^3 + 12*q^4 - 8*q^5 - 12*q^6 + 30*q^7 + ...
%t a[ n_] := SeriesCoefficient[ -5 + 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 *)
%t QP = QPochhammer; s = -5*q +QP[q^2]^14/(QP[q]^5*QP[q^3]*QP[q^4]^5* QP[q^6]^2*QP[q^12]) + O[q]^50; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 16 2015, adapted from PARI *)
%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A =x * O(x^n); polcoeff( -5 * x + 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))}; /* _Michael Somos_, May 20 2015 */
%Y Cf. A007258, A187146, A187147, A187148.
%K sign
%O -1,3
%A _Michael Somos_, Aug 28 2005