OFFSET
0,2
COMMENTS
a(n) = 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 2-colored horizontal steps H(k) = (k,0) for every positive integer k.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
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-2*n)} (C(n)*binomial(m+2*n,m) *binomial(s-2*n-1,m-1) * 2^m)), where C(n) = A000108(n).
G.f.: (1-3*z-sqrt(1-6*z+5*z^2+8*z^3-4*z^4))/(2*z^2*(1-z)).
a(n) ~ sqrt(221 + 53*sqrt(17)) * (5+sqrt(17))^n / (sqrt(Pi) * n^(3/2) * 2^(n+2)). - Vaclav Kotesovec, Apr 20 2015
Recurrence: (n+2)*a(n) = (7*n+5)*a(n-1) - (11*n-2)*a(n-2) - 3*(n-5)*a(n-3) + 12*(n-3)*a(n-4) - 4*(n-4)*a(n-5). - Vaclav Kotesovec, Apr 20 2015
MATHEMATICA
CoefficientList[Series[(1-3*x-Sqrt[1-6*x+5*x^2+8*x^3-4*x^4])/(2*x^2*(1-x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 20 2015 *)
PROG
(PARI) x='x+O('x^50); Vec((1-3*x-sqrt(1 -6*x +5*x^2 +8*x^3 -4*x^4) )/(2*x^2*(1-x))) \\ G. C. Greubel, Jun 03 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
José Luis Ramírez Ramírez, Apr 19 2015
STATUS
approved