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A285043
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Expansion of cosh(3*arctanh(2*sqrt(x))).
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5
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1, 18, 102, 500, 2310, 10332, 45276, 195624, 836550, 3549260, 14965236, 62783448, 262303132, 1092063000, 4533175800, 18769219920, 77539370310, 319704052140, 1315894618500, 5407825361400, 22193291140020
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OFFSET
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0,2
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COMMENTS
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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.
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.
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LINKS
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FORMULA
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a(n) = (8*n + 1)*binomial(2*n,n).
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 + ....
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
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MAPLE
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seq((8*n + 1)*binomial(2*n, n), n = 0..20);
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MATHEMATICA
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CoefficientList[Series[Cosh[3*ArcTanh[2*Sqrt[x]]], {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 10 2017 *)
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PROG
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(PARI) x='x + O('x^30); Vec((1 + 12*x)/(1 - 4*x)^(3/2)) \\ Indranil Ghosh, Apr 10 2017
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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