OFFSET
0,2
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,i), where C(n) 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*(1-3*z)).
a(n) ~ sqrt(5) * 4^(n+1) / (sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 21 2015
Conjecture: (n+2)*a(n) +6*(-n-1)*a(n-1) +(5*n+4)*a(n-2) +6*(2*n-3)*a(n-3)=0. - R. J. Mathar, Sep 24 2016
G.f. A(x) satisfies: A(x) = 1/(1 - 3*x) + x^2 * A(x)^2. - Ilya Gutkovskiy, Jun 30 2020
EXAMPLE
For n=2 we have 10 paths: H(1)H(1), H(1)H(2), H(1)H(3), H(2)H(1), H(2)H(2), H(2)H(3), H(3)H(1), H(3)H(2), H(3)H(3) and UD.
MATHEMATICA
CoefficientList[Series[(1-3*x-Sqrt[(1-3*x)*(1-3*x-4x^2)])/(2*x^2*(1-3*x)), {x, 0, 20}], x] (* Vaclav Kotesovec, Apr 21 2015 *)
PROG
(PARI) Vec((1-3*x-sqrt((1-3*x)*(1-3*x-4*x^2)))/(2*x^2*(1-3*x)) + O(x^50)) \\ G. C. Greubel, Feb 05 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
José Luis Ramírez Ramírez, Apr 20 2015
STATUS
approved