A127846 Series reversion of x/(1+5x+4x^2).

0, 1, 5, 29, 185, 1257, 8925, 65445, 491825, 3768209, 29324405, 231153133, 1841801065, 14810069497, 120029657805, 979470140661, 8040831465825, 66361595715105, 550284185213925, 4582462506008253, 38306388126997785

Offset: 0

Author

Paul Barry, Feb 02 2007

Keywords

• easy
• nonn

Comments

Hankel transform is -A127847(n)=-4^C(n,2)*(4^n-1)/3; a(n+1) counts (5,4)-Motzkin paths of length n, where there are 5 colors available for the H(1,0) steps and 4 for the U(1,1) steps. See A059231 for more information.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A059231

Programs

Mathematica

CoefficientList[Series[(1-5*x-Sqrt[1-10*x+9*x^2])/(8*x), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 19 2012 *)

Sage

A127846 = lambda n: hypergeometric([1-n, -n], [2], 4) if n>0 else 0
[Integer(A127846(n).n(100)) for n in (0..22)] # Peter Luschny, Sep 23 2014

Formula

G.f.: (1-5x-sqrt(1-10x+9x^2))/(8x); a(n)=sum{k=0..n-1, (1/n)*C(n,k)C(n,k+1)4^k}; a(n+1)=sum{k=0..floor(n/2), C(n, 2k)C(k)5^(n-2k)*4^k};

Recurrence: (n+1)*a(n) = 5*(2*n-1)*a(n-1) - 9*(n-2)*a(n-2). - Vaclav Kotesovec, Oct 19 2012

a(n) ~ 3^(2*n+1)/(4*sqrt(2*Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 19 2012

a(n) = A059231(n) for n>0. - Philippe Deléham, Apr 03 2013

a(n) = hypergeom([1-n, -n], [2], 4) for n>0. - Peter Luschny, Sep 23 2014