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%I #14 Jan 30 2020 21:29:18
%S 1,18,102,500,2310,10332,45276,195624,836550,3549260,14965236,
%T 62783448,262303132,1092063000,4533175800,18769219920,77539370310,
%U 319704052140,1315894618500,5407825361400,22193291140020
%N Expansion of cosh(3*arctanh(2*sqrt(x))).
%C Note that the function cosh(2*n*arctanh(sqrt(x)) is the o.g.f. for the coordination sequence of the C_n lattice. See, for example, A010006.
%C In A285043 through A285046 we consider sequences with o.g.f. cosh((2*n+1)*arctanh(2*sqrt(x)) for n = 1, 2, 3 and 4. For n = 0 we get the central binomial coefficients A000984.
%F a(n) = (8*n + 1)*binomial(2*n,n).
%F O.g.f. cosh(3*arctanh(2*sqrt(x))) = (1 + 12*x)/(1 - 4*x)^(3/2) = 1 + 18*x + 102*x^2 + 500*x^3 + ....
%F D-finite with recurrence: n*a(n) +2*(4*n-13)*a(n-1) +24*(-2*n+3)*a(n-2)=0. - _R. J. Mathar_, Jan 22 2020
%p seq((8*n + 1)*binomial(2*n,n), n = 0..20);
%t CoefficientList[Series[Cosh[3*ArcTanh[2*Sqrt[x]]], {x, 0, 20}], x] (* _Vaclav Kotesovec_, Apr 10 2017 *)
%o (PARI) x='x + O('x^30); Vec((1 + 12*x)/(1 - 4*x)^(3/2)) \\ _Indranil Ghosh_, Apr 10 2017
%Y Cf. A000984, A010006, A085840, A285044, A285045, A285046.
%K nonn,easy
%O 0,2
%A _Peter Bala_, Apr 09 2017
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