A285044 Expansion of cosh(5*arctanh(2*sqrt(x))).

1, 50, 550, 4020, 24710, 138012, 725340, 3655080, 17859270, 85230860, 399257716, 1842353240, 8396404380, 37868584600, 169278679800, 750923914320, 3308947546950, 14495583969900, 63172016823300, 274031830241400, 1183780040663220

Offset: 0

Author

Peter Bala, Apr 10 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: n = 0 gives the central binomial coefficients A000984.

Crossrefs

Cf. A000984, A010006, A085840, A285043, A285045, A285046.

Programs

Maple

seq(1/3*(64*n^2 + 8*n + 3)*binomial(2*n,n), n = 0..20);

Formula

a(n) = 1/3*(64*n^2 + 8*n + 3)*binomial(2*n,n).

O.g.f. cosh(5*arctanh(2*sqrt(x))) = (1 + 40*x + 80*x^2)/(1 - 4*x)^(5/2) = 1 + 50*x + 550*x^2 + 4020*x^3 + ....

Note that the zeros of the polynomial 1 + 40*x^2 + 80*x^4 = 1/2*((1 + 2*x)^5 + (1 - 2*x)^5), are given by 1/2*cot(k*Pi/5)*i for 1 <= k <= 4. See A085840.

O.g.f. for the sequence with interpolated zeros: 1/2*( ((1 + 2*x)/(1 - 2*x))^(5/2) + ((1 - 2*x)/(1 + 2*x))^(5/2) ) = 1 + 50*x^2 + 550*x^4 + 4020*x^6 + ....