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%I #32 Mar 12 2021 22:24:46
%S 1,4,14,36,85,180,360,684,1246,2196,3754,6264,10226,16380,25804,40032,
%T 61275,92628,138452,204804,300040,435672,627356,896400,1271525,
%U 1791324,2507426,3488472,4825531,6638688,9085888,12373992,16772908,22633812,30411780,40695048
%N McKay-Thompson series of class 12H for the Monster group with a(0) = 4.
%C The reason for having multiple versions of these McKay-Thompson series is that each one is a quotient of Dedekind eta-functions, differing only in their constant terms. Each one is equally important.
%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).
%H G. C. Greubel, <a href="/A187091/b187091.txt">Table of n, a(n) for n = -1..1000</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 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 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 c(q) * b(q^4) / (b(q) * c(q^4)) in powers of q where b(), c() are cubic AGM theta functions.
%F Expansion of (1/q) * ((chi(-q^3) * chi(-q^6)) / (chi(-q) * chi(-q^3)))^4 in powers of q where chi() is a Ramanujan theta function.
%F Expansion of (eta(q^3) * eta(q^4) / (eta(q) * eta(q^12)))^4 in powers of q.
%F Euler transform of period 12 sequence [ 4, 4, 0, 0, 4, 0, 4, 0, 0, 4, 4, 0, ...].
%F G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = f(t) where q = exp(2 Pi i t).
%F Convolution square of A058578. a(n) = A058486(n) unless n=0.
%F a(n) = -(-1)^n * A193522(n). - _Michael Somos_, Sep 05 2015
%F a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Sep 10 2015
%e G.f. = 1/q + 4 + 14*q + 36*q^2 + 85*q^3 + 180*q^4 + 360*q^5 + 684*q^6 + 1246*q^7 + ...
%t a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ q^3] QPochhammer[ q^4] / (QPochhammer[ q] QPochhammer[ q^12]))^4, {q, 0, n}]; (* _Michael Somos_, Sep 05 2015 *)
%t a[ n_] := SeriesCoefficient[ 1/q (QPochhammer[ -q, q] QPochhammer[ -q^2, q^2] QPochhammer[ q^3, q^6] QPochhammer[ q^6, q^12])^4, {q, 0, n}]; (* _Michael Somos_, Sep 05 2015 *)
%t nmax = 50; CoefficientList[Series[Product[((1-x^(3*k)) * (1-x^(4*k)) / ((1-x^k) * (1-x^(12*k))))^4, {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Sep 10 2015 *)
%o (PARI) {a(n) = my(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( (eta(x^3 + A) * eta(x^4 + A) / (eta(x + A) * eta(x^12 + A)))^4, n))};
%Y Cf. A058486, A058578, A112192, A193522.
%K nonn
%O -1,2
%A _Michael Somos_, Mar 06 2011
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