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 A062246 McKay-Thompson series of class 27c for the Monster group. 5

%I

%S 1,-1,-1,0,0,1,0,1,0,1,-1,-1,-1,0,1,-1,1,0,2,-2,-2,-1,1,2,-1,2,1,3,-3,

%T -3,-2,1,3,-2,3,0,5,-5,-5,-3,1,5,-3,5,1,7,-7,-7,-5,2,7,-4,7,1,11,-11,

%U -11,-6,3,11,-6,11,2,15,-15,-15,-10,4,15,-9,14,2,22,-22,-22,-13,6,21,-12,21,4,30,-30,-30,-19,8,29,-17,28,4,42

%N McKay-Thompson series of class 27c for the Monster group.

%H Seiichi Manyama, <a href="/A062246/b062246.txt">Table of n, a(n) for n = 0..10000</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 J. McKay and A. Sebbar, <a href="http://dx.doi.org/10.1007/s002080000116">Fuchsian groups, automorphic functions and Schwarzians</a>, Math. Ann., 318 (2000), 255-275.

%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>

%F Expansion of q^(1/3) * eta(q) / eta(q^9) in powers of q.

%F Euler transform of period 9 sequence [ -1, -1, -1, -1, -1, -1, -1, -1, 0, ...].

%F a(n) = (-1)^n * A062245(n).

%F a(n) = -(1/n)*Sum_{k=1..n} A116607(k)*a(n-k), a(0) = 1. - _Seiichi Manyama_, Mar 25 2017

%e 1 - x - x^2 + x^5 + x^7 + x^9 - x^10 - x^11 - x^12 + x^14 - x^15 + x^16 + ...

%e T27c = 1/q - q^2 - q^5 + q^14 + q^20 + q^26 - q^29 - q^32 - q^35 + q^41 - ...

%t QP = QPochhammer; s = QP[q]/QP[q^9] + O[q]^90; CoefficientList[s, q] (* _Jean-François Alcover_, Nov 12 2015 *)

%o (PARI) {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) / eta(x^9 + A), n))} /* _Michael Somos_, Jun 26 2004 */

%Y Cf. A007706, A062245.

%K sign

%O 0,19

%A _N. J. A. Sloane_, Jul 01 2001