A026327 a(n) = 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) = 4. Also a(n) = T(n,n-2), where T is the array in A026323.

1, 3, 10, 30, 90, 266, 783, 2295, 6710, 19580, 57057, 166101, 483210, 1405080, 4084590, 11872494, 34508997, 100313635, 291646580, 848102640, 2466916474, 7177785582, 20891443950, 60827142350, 177167486925, 516217883571, 1504692189588

Offset: 2

Author

Clark Kimberling

Keywords

• nonn

Comments

Number of paths in the plane x>=0 and y>=-2, from (0,0) to (n,2), and consisting of steps U=(1,1), D=(1,-1) and H=(1,0). For example, for n=4, we have the 10 paths: UUUD, UUHH, UUDU, UHUH, UHHU, UDUU, HUUH, HUHU, HHUU, DUUU. - José Luis Ramírez Ramírez, Apr 20 2015

Links

• G. C. Greubel, Table of n, a(n) for n = 2..1000

Programs

Mathematica

Rest[Rest[CoefficientList[Series[x^2*((1-x-Sqrt[1-2*x-3*x^2])/(2*x^2))^2/(1-x-x^2*((1-x-Sqrt[1-2*x-3*x^2])/(2*x^2)+1/(1-x-x^2/(1-x)))), {x, 0, 20}], x]]] (* Vaclav Kotesovec, Apr 21 2015 *)

Formula

Conjecture: -(n+8)*(n-2)*a(n) +3*(2*n^2+7*n-28)*a(n-1) +3*(-3*n^2-n+36)*a(n-2) -4*(n+4)*(n-1)*a(n-3) +12*(n-1)*(n-2)*a(n-4)=0. - R. J. Mathar, Jun 23 2013

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

a(n) ~ 5 * 3^(n+5/2) / (2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 21 2015