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%I #25 Mar 12 2021 22:24:46
%S 1,-4,5,0,-5,0,9,0,-14,0,19,0,-34,0,55,0,-69,0,104,0,-164,0,209,0,
%T -283,0,413,0,-539,0,712,0,-968,0,1248,0,-1642,0,2167,0,-2731,0,3526,
%U 0,-4592,0,5736,0,-7244,0,9255,0,-11520,0,14378,0,-18018,0,22238,0,-27556,0
%N McKay-Thompson series of class 12c for the Monster group with a(0) = -4.
%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
%D D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).
%H Seiichi Manyama, <a href="/A186930/b186930.txt">Table of n, a(n) for n = -1..10000</a>
%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 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>
%F Expansion of (b(q) * c(q^2)^3) / (c(q) * c(q^4)^2 * b(q^4)) in powers of q where b(), c() are cubic AGM functions.
%F Expansion of (1/q) * chi(q) * chi(-q)^5 * chi(q^3)^5 * chi(-q^3) in powers of q where chi() is a Ramanujan theta function.
%F Expansion of (eta(q)^4 * eta(q^6)^9) / (eta(q^2)^3 * eta(q^3)^4 * eta(q^4) * eta(q^12)^5) in powers of q.
%F Euler transform of period 12 sequence [ -4, -1, 0, 0, -4, -6, -4, 0, 0, -1, -4, 0, ...].
%F a(2*n) = 0 unless n=0. a(2*n - 1) = A058491(n).
%F a(n) = A187045(n) unless n=0. - _Michael Somos_, Sep 05 2015
%e G.f. = 1/q - 4 + 5*q - 5*q^3 + 9*q^5 - 14*q^7 + 19*q^9 - 34*q^11 + 55*q^13 + ...
%t a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q, q^2] QPochhammer[ -q^3, q^6])^5 (QPochhammer[ -q, q^2] QPochhammer[ q^3, q^6]), {q, 0, n}]; (* _Michael Somos_, Sep 05 2015 *)
%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x + A)^4 * eta(x^6 + A)^9) / (eta(x^2 + A)^3 * eta(x^3 + A)^4 * eta(x^4 + A) * eta(x^12 + A)^5), n))};
%Y Cf. A058491, A187045.
%K sign
%O -1,2
%A _Michael Somos_, Mar 07 2011
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