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%I #26 Jun 21 2018 03:07:15
%S 1,0,1,-2,0,2,1,0,3,2,0,2,-4,0,1,0,0,2,7,0,4,-10,0,8,3,0,8,10,0,8,-15,
%T 0,7,2,0,10,22,0,17,-32,0,22,10,0,26,32,0,24,-48,0,25,8,0,30,62,0,43,
%U -88,0,58,22,0,65,88,0,66,-127,0,66,22,0,80,152,0,107,-214,0,136,52,0
%N McKay-Thompson series of class 18j for the Monster group.
%C G.f. A(x) satisfies: 0 = f(A(x), A(x^2)) = f(A(x), A(-x)) where f(u, v) = 32 + 4 * (u + v) - 2 * (u^2 + v^2) + 2 * (u^3 + v^3) - 3*u*v * (u + v) + (u^4 + v^4) + u*v * (u^2 + v^2) - (u*v)^2 * (u + v). - _Michael Somos_, Apr 20 2004
%H G. C. Greubel, <a href="/A058548/b058548.txt">Table of n, a(n) for n = -1..2500</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 <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>
%F a(3*n) = 0.
%F Expansion of A + 1/A, where A = (eta(q^3)*eta(q^18)^2*eta(q^27)/(eta(q^6) *eta(q^9)^2*eta(q^54)))^2, in powers of q. - _G. C. Greubel_, Jun 21 2018
%e T18j = 1/q + q - 2*q^2 + 2*q^4 + q^5 + 3*q^7 + 2*q^8 + 2*q^10 - 4*q^11 + ...
%t nmax = 80; QP = QPochhammer; A = x^2*O[x]^nmax; A = ((QP[A + x^3]*QP[A + x^18]^2*QP[A + x^27])/(QP[A + x^6]*QP[A + x^9]^2*QP[A + x^54]))^2/x; a[n_] := SeriesCoefficient[A + 1/A, n]; Table[a[n], {n, -1, nmax}] (* _Jean-François Alcover_, Nov 14 2015, adapted from PARI *)
%t eta[q_] := q^(1/24)*QPochhammer[q]; A := q*(eta[q^3]*eta[q^18]^2* eta[q^27]/( eta[q^6]*eta[q^9]^2*eta[q^54]))^2; a := CoefficientList[ Series[A + q^2/A, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* _G. C. Greubel_, Jun 21 2018 *)
%o (PARI) {a(n) = local(A); if( n<0, n==-1, A = x^2 * O(x^n); A = ((eta(x^3 + A) * eta(x^18 + A)^2 * eta(x^27 + A)) / (eta(x^6 + A) * eta(x^9 + A)^2 * eta(x^54 + A)))^2 / x; polcoeff( A + 1/A, n))} /* _Michael Somos_, Apr 20 2004 */
%Y Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.
%K sign
%O -1,4
%A _N. J. A. Sloane_, Nov 27 2000
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