A285043 Expansion of cosh(3*arctanh(2*sqrt(x))).

1, 18, 102, 500, 2310, 10332, 45276, 195624, 836550, 3549260, 14965236, 62783448, 262303132, 1092063000, 4533175800, 18769219920, 77539370310, 319704052140, 1315894618500, 5407825361400, 22193291140020

Offset: 0

Author

Peter Bala, Apr 09 2017

Keywords

• nonn
• easy

Comments

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.

Crossrefs

Cf. A000984, A010006, A085840, A285044, A285045, A285046.

Programs

Maple

seq((8*n + 1)*binomial(2*n,n), n = 0..20);

Mathematica

CoefficientList[Series[Cosh[3*ArcTanh[2*Sqrt[x]]], {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 10 2017 *)

PARI

my(x='x + O('x^30)); Vec((1 + 12*x)/(1 - 4*x)^(3/2)) \\ Indranil Ghosh, Apr 10 2017

Formula

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