OFFSET
0,2
COMMENTS
This generating function, together with the multiplier function -xg(x), produce an involution in the Riordan group.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Naiomi T. Cameron and Asamoah Nkwanta, On Some (Pseudo) Involutions in the Riordan Group, Journal of Integer Sequences, Vol. 8 (2005), Article 05.3.7.
FORMULA
G.f.: (g(x)*(1+x))/(1-x*g(x)) where g(x)=((1-x+x^2)-sqrt((1-x+x^2)^2-4x^2))/(2*x^2).
Conjecture: -(n+1)*(n-2)*a(n) +2*(n^2-n-3)*a(n-1) +(n^2-3*n+8)*a(n-2) +2*(n^2-5*n+3)*a(n-3) -(n-1)*(n-4)*a(n-4)=0. - R. J. Mathar, Jun 17 2016
a(n) ~ 5^(1/4) * phi^(2*n+1) / sqrt(Pi*n), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, May 29 2022
EXAMPLE
a(3) = 13 since there are 12 = 3*2*2 paths that stay at level 0 and one path ULD that goes above level 0.
MATHEMATICA
CoefficientList[Series[(((1 - x + x^2) - Sqrt[(1 - x + x^2)^2 - 4 x^2])/(2*x^2)*(1 + x))/(1 - x*((1 - x + x^2) - Sqrt[(1 - x + x^2)^2 - 4 x^2])/(2*x^2)), {x, 0, 30}], x] (* Wesley Ivan Hurt, Jan 10 2017 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Louis Shapiro and Gi-Sang Cheon, Jan 15 2007
STATUS
approved