A026325 Number of paths in the plane x >= 0 and y >= -2, from (0,0) to (n,0), and consisting of steps U = (1,1), D = (1,-1) and H = (1,0).

1, 1, 3, 7, 19, 51, 140, 386, 1071, 2983, 8338, 23376, 65715, 185199, 523134, 1480872, 4200411, 11936619, 33981063, 96897759, 276739029, 791532973, 2267119660, 6502108902, 18671460905, 53680763201, 154507444731, 445190930863, 1284064525987

Offset: 0

Author

Clark Kimberling

Keywords

• nonn

Comments

Previous name: Number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| <= 1 for i = 1,2,...,n, s(0) = 2, s(n) = 2. Also T(n,n), where T is the array in A026323.

Number of paths in the plane x >= 0 and y >= -2, from (0,0) to (n,0), and consisting of steps U = (1,1), D = (1,-1) and H = (1,0). For example, for n=3, we have the 7 paths: HHH, UDH, HUD, UHD, HDU, DUH, DHU. - José Luis Ramírez Ramírez, Apr 20 2015

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A001006, A026327.

Programs

Maple

gf := sqrt(4 - 8*x - 12*x^2)*(1/x^5 - 1/x^4 - 1/(4*x^6)):
ser := series(gf,x,36): seq(coeff(ser, x, n), n=0..28);
A026325 := proc(n) option remember; ifelse(n < 3, [1, 1, 3][n + 1],
((4*n+15)*A026325(n-1) + (3-n)*A026325(n-2) - 6*n*A026325(n-3))/(n+6)) end:
seq(A026325(n), n = 0..28); # Peter Luschny, Oct 06 2022

Mathematica

CoefficientList[Series[1/(1 - x - x^2 ((1 - x - (1 - 2 x - 3 x^2)^(1/2))/(2 x^2) + 1/(1 - x - x^2 / (1 - x)))), {x, 0, 40}], x] (* Vincenzo Librandi, Apr 21 2015 *)

PARI

x='x+O('x^50); Vec(1/(1 - x - x^2*((1 - x - (1 - 2*x - 3*x^2)^(1/2))/(2*x^2) + 1/(1 - x - x^2/(1 - x))))) \\ G. C. Greubel, Feb 15 2017

Formula

G.f: 1/(1 - x - x^2*(M(x) + 1/(1 - x - x^2/(1 - x)))), where M(x) is g.f. of Motzkin paths A001006. - José Luis Ramírez Ramírez, Apr 20 2015

a(n) ~ 3^(n + 7/2)/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 21 2015

(n + 6)*a(n) + (-4*n - 15)*a(n-1) + (n - 3)*a(n-2) + 6*n*a(n-3) = 0. - R. J. Mathar, Jul 23 2017

Extensions

New name using a comment of José Luis Ramírez Ramírez by Peter Luschny, Oct 06 2022