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 A187144 McKay-Thompson series of class 12I for the Monster group with a(0) = 1. 0

%I

%S 1,1,2,0,1,0,0,0,-2,0,-2,0,2,0,4,0,3,0,-4,0,-8,0,-4,0,5,0,14,0,7,0,-8,

%T 0,-20,0,-12,0,14,0,28,0,17,0,-20,0,-44,0,-24,0,28,0,66,0,36,0,-40,0,

%U -90,0,-52,0,56,0,124,0,71,0,-80,0,-176,0,-96,0,109,0,244,0,133,0,-144

%N McKay-Thompson series of class 12I for the Monster group with a(0) = 1.

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

%H J. M. Borwein and P. B. Borwein, <a href="http://dx.doi.org/10.1090/S0002-9947-1991-1010408-0">A cubic counterpart of Jacobi's identity and the AGM</a>, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701.

%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).

%F Expansion of c(q) / c(q^4) in powers of q where c() is a cubic AGM function.

%F Expansion of eta(q^3)^3 * eta(q^4) / (eta(q) * eta(q^12)^3) in powers of q.

%F Euler transform of period 12 sequence [ 1, 1, -2, 0, 1, -2, 1, 0, -2, 1, 1, 0, ...].

%F Convolution inverse of A123649.

%F a(2*n) = 0 unless n=0. a(2*n - 1) = A058487(n).

%e G.f. = 1/q + 1 + 2*q + q^3 - 2*q^7 - 2*q^9 + 2*q^11 + 4*q^13 + 3*q^15 - 4*q^17 + ...

%t QP = QPochhammer; s = QP[q^3]^3*(QP[q^4]/(QP[q]*QP[q^12]^3)) + O[q]^80; 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( eta(x^3 + A)^3 * eta(x^4 + A) / (eta(x + A) * eta(x^12 + A)^3), n))};

%Y Cf. A058487, A123649.

%K sign

%O -1,3

%A _Michael Somos_, Mar 05 2011

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