OFFSET
2,2
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
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
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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved