OFFSET
0,2
COMMENTS
Number of lattice paths, never going below the x-axis, from (0,0) to (n,0) consisting of up steps U = (1,1), down steps D = (1,-1) and 3-colored horizontal steps H(k) = (k,0) for every positive integer k.
LINKS
R. De Castro, A. L. Ramírez and J. L. Ramírez, Applications in Enumerative Combinatorics of Infinite Weighted Automata and Graphs, Scientific Annals of Computer Science, 24(1)(2014), 137-171.
FORMULA
a(s) = Sum_{n=0..s} (Sum_{m=0..s-2n} (C(n)binomial(m+2n,m)*binomial(s-2n-1,m-1)3^m)), where C(n)=A000108(n).
G.f.: (1-4z-sqrt(1-8z+12z^2+8z^3-4z^4))/(2z^2(1-z)).
a(n) ~ sqrt(77 + 29*sqrt(7)) * (3+sqrt(7))^n / (sqrt(3*Pi) * n^(3/2)). - Vaclav Kotesovec, Apr 20 2015
Recurrence: (n+2)*a(n) = 3*(3*n+2)*a(n-1) - 4*(5*n-2)*a(n-2) + 4*(n+2)*a(n-3) + 12*(n-3)*a(n-4) - 4*(n-4)*a(n-5). - Vaclav Kotesovec, Apr 20 2015
MATHEMATICA
CoefficientList[Series[(1-4*x-Sqrt[1-8*x+12*x^2+8*x^3-4*x^4])/(2*x^2*(1-x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 20 2015 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
José Luis Ramírez Ramírez, Apr 19 2015
STATUS
approved