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 A112146 McKay-Thompson series of class 9b for the Monster group. 5

%I

%S 1,0,9,-4,0,36,2,0,126,12,0,324,-21,0,801,4,0,1764,36,0,3744,-68,0,

%T 7452,21,0,14400,112,0,26748,-184,0,48510,44,0,85536,275,0,147924,

%U -456,0,250452,112,0,417276,644,0,683640,-1019,0,1104948,240,0,1761552,1370,0

%N McKay-Thompson series of class 9b for the Monster group.

%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

%H G. A. Edgar, <a href="/A112146/b112146.txt">Table of n, a(n) for n = -1..999</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 Expansion of q^(1/3) * 3*( b(q) / c(q) + c(q) / b(q)) in powers of q where b(), c() are cubic AGM theta functions. - _Michael Somos_, Mar 24 2007

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u+v)^3 - (u^2 + 3*u - 18) * (v^2 + 3*v - 18).

%F G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 + w^2 + u*w + 18*(u+w) - (w+u)*v^2 - 9*v + 54.

%F Expansion of ( (eta(q^3) / eta(q^9))^4 + 9 * (eta(q^9) / eta(q^3))^4) in powers of q.

%F a(3*n) = 0. a(3*n-1) = A058095(n). a(3*n + 1) = 9 * A128758(n). - _Michael Somos_, Feb 19 2015

%e T9b = 1/q + 9*q - 4*q^2 + 36*q^4 + 2*q^5 + 126*q^7 + 12*q^8 + 324*q^10 + ...

%t QP = QPochhammer; s = (QP[q^3]^8 + 9*q^2*QP[q^9]^8)/(QP[q^3]^4*QP[q^9]^4) + O[q]^60; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 14 2015, adapted from PARI *)

%o (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); A = (eta(x^3 + A) / eta(x^9 + A))^4; polcoeff( A + 9*x^2 / A, n))}; /* _Michael Somos_, Mar 24 2007 */

%Y Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.

%Y Cf. A058095, A128758.

%K sign

%O -1,3

%A _Michael Somos_, Aug 28 2005, Aug 09 2008

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Last modified January 16 06:59 EST 2019. Contains 319188 sequences. (Running on oeis4.)