A091148 Expansion of (1-x-sqrt(1-2x-19x^2))/(10x^2).

1, 1, 6, 16, 81, 301, 1451, 6231, 29891, 137731, 666976, 3193026, 15658831, 76719891, 380788006, 1894818776, 9502977851, 47822585931, 241944876266, 1228151169656, 6258922649451, 31992657321551, 164040821525031

Offset: 0

Author

Paul Barry, Dec 22 2003

Keywords

• easy
• nonn

Comments

Number of lattice paths in the first quadrant from (0,0) to (n,0) using only steps H=(1,0), U=(1,1) and D=(1,-1), where the U steps come in 5 colors (i.e. Motzkin paths with the up steps in 5 colors). Series reversion of x/(1+x+5x^2). - Paul Barry, May 16 2005

Links

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

Crossrefs

Cf. A217275.

Programs

Mathematica

CoefficientList[Series[(1 - x - Sqrt[1 - 2 x - 19 x^2]) / (10 x^2), {x, 0, 30}], x] (* Vincenzo Librandi, May 10 2013 *)

PARI

x='x+O('x^66); Vec((1-x-sqrt(1-2*x-19*x^2))/(10*x^2)) \\ Joerg Arndt, May 11 2013

Formula

a(n) = A014434(n+1)/5.

G.f.: 2/(1-x+sqrt(1-2x-19x^2)).

a(n) = sum{k=0..n, binomial(n, k)5^(k/2)C(k/2)(1+(-1)^k)/2}, C(n)=A000108(n).

a(n) = sum{k=0..n, C(n, 2k)C(k)5^k}; - Paul Barry, May 16 2005

D-finite with recurrence: (n+2)*a(n) -(2*n+1)*a(n-1) +19*(1-n)*a(n-2)=0. - R. J. Mathar, Sep 26 2012

a(n) ~ 1/10*sqrt(230+61*sqrt(5))/(n^(3/2)*sqrt(Pi))*(1+2*sqrt(5))^n. - Vaclav Kotesovec, Sep 29 2012

G.f.: 1/(1 - x - 5*x^2/(1 - x - 5*x^2/(1 - x - 5*x^2/(1 - x - 5*x^2/(1 - ....))))), a continued fraction. - Ilya Gutkovskiy, May 26 2017