OFFSET
0,4
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
a(n) = Sum_{i=0..floor(n/2)} 3^(n-2i)*C(i)*binomial(n-i-1,n), where C(i) is the n-th Catalan number A000108.
G.f.: (1 - 3*z - sqrt((1-3*z)*(1-3*z-4*z^2)))/(2*z^2).
a(n) ~ sqrt(5) * 4^n / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 21 2015
Conjecture: (n+2)*a(n) +3*(-2*n-1)*a(n-1) +5*(n-1)*a(n-2) +6*(2*n-5)*a(n-3)=0. - R. J. Mathar, Sep 24 2016
EXAMPLE
For n=3 we have 3 paths: UH1D, UH2D, UH3D.
MATHEMATICA
CoefficientList[Series[(1-3*x-Sqrt[(1-3*x)*(1-3*x-4*x^2)])/(2*x^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 21 2015 *)
PROG
(PARI) x='x+O('x^50); Vec((1-3*x-sqrt((1-3*x)*(1-3*x-4*x^2)))/(2*x^2)) \\ G. C. Greubel, Feb 14 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
José Luis Ramírez Ramírez, Apr 20 2015
STATUS
approved