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A127846 Series reversion of x/(1+5x+4x^2). 4
0, 1, 5, 29, 185, 1257, 8925, 65445, 491825, 3768209, 29324405, 231153133, 1841801065, 14810069497, 120029657805, 979470140661, 8040831465825, 66361595715105, 550284185213925, 4582462506008253, 38306388126997785 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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

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

MATHEMATICA

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

PROG

(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

CROSSREFS

Cf. A059231

Sequence in context: A153391 A175891 A081336 * A059231 A137573 A234317

Adjacent sequences:  A127843 A127844 A127845 * A127847 A127848 A127849

KEYWORD

easy,nonn

AUTHOR

Paul Barry, Feb 02 2007

STATUS

approved

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Last modified February 20 15:20 EST 2019. Contains 320336 sequences. (Running on oeis4.)