A256939 Expansion of g.f.: (1-4*z-sqrt(1-8*z+12*z^2+8*z^3-4*z^4))/(2*z^2(1-z)).

1, 3, 13, 57, 257, 1185, 5573, 26661, 129437, 636429, 3163725, 15877101, 80340813, 409495053, 2100558429, 10836262173, 56184433661, 292628726205, 1530338756093, 8032671187581, 42304703640701, 223484135199357, 1183921500416509, 6288098247289341

Offset: 0

Author

José Luis Ramírez Ramírez, Apr 19 2015

Keywords

• nonn

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.

Crossrefs

Cf. A135052.

Programs

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 *)

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