A127360 a(n) = Sum_{k=0..n} binomial(n, floor(k/2))*4^(n-k).

1, 5, 22, 95, 406, 1730, 7360, 31295, 133030, 565430, 2403172, 10213670, 43408444, 184486580, 784069252, 3332296895, 14162266630, 60189642830, 255806000260, 1087175537570, 4620496103956, 19637108580380, 83457711731152, 354695275386470, 1507454921406556

Offset: 0

Author

Paul Barry, Jan 11 2007

Keywords

• easy
• nonn

Comments

Hankel transform is (-3)^n. In general, given r >= 0, the sequence given by Sum_{k=0..n} binomial(n, floor(k/2))*r^(n-k) has Hankel transform (1-r)^n. The sequence is the image of the sequence with g.f. (1+x)/(1-4x) under the Chebyshev mapping g(x)->(1/sqrt(1-4x^2))g(xc(x^2)), where c(x) is the g.f. of the Catalan numbers A000108.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..300
• Isaac DeJager, Madeleine Naquin, Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.

Crossrefs

Cf. A107430. - Philippe Deléham, Sep 16 2009

Cf. A061554.

Programs

Mathematica

CoefficientList[Series[(1/Sqrt[1-4x^2])*(1+x*(1-Sqrt[1-4*x^2])/(2*x^2))/(1-4*x*(1-Sqrt[1-4*x^2])/(2*x^2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 19 2012 *)

Formula

G.f.: (1/sqrt(1-4*x^2))*(1+x*c(x^2))/(1-4*x*c(x^2)) with c(x) = (1-sqrt(1-4*x))/(2*x).

a(n) = Sum_{k=0..n} A061554(n,k)*4^k. - Philippe Deléham, Dec 04 2009

Recurrence: 4*n*a(n) = (17*n + 8)*a(n-1) + 2*(8*n - 33)*a(n-2) - 68*(n-2)*a(n-3). - Vaclav Kotesovec, Oct 19 2012

a(n) ~ 5*17^n/4^(n+1). - Vaclav Kotesovec, Oct 19 2012