A256938 Expansion of g.f.: (1-3*z-sqrt(1-6*z+5*z^2+8*z^3-4*z^4))/(2*z^2*(1-z)).

1, 2, 7, 24, 86, 316, 1189, 4562, 17796, 70398, 281812, 1139658, 4649402, 19112962, 79096155, 329258424, 1377798890, 5792421108, 24454224310, 103631241912, 440674939192, 1879769835968, 8041447249926, 34490981798188, 148295899087660, 639036278210420

Offset: 0

Author

José Luis Ramírez Ramírez, Apr 19 2015

Keywords

• nonn

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

Crossrefs

Cf. A135052.

Programs

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 *)

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

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