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A026325 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. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

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

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

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 *)

PROG

(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

CROSSREFS

Cf. A026327.

Sequence in context: A078059 A018031 A052948 * A002426 A011769 A087432

Adjacent sequences:  A026322 A026323 A026324 * A026326 A026327 A026328

KEYWORD

nonn

AUTHOR

Clark Kimberling

STATUS

approved

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Last modified October 22 10:38 EDT 2018. Contains 316436 sequences. (Running on oeis4.)