OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1000
Markus Kuba, A note on naturally embedded ternary trees, Electronic Journal of Combinatorics, Volume 18 (1), paper P142, 2011.
Markus Kuba, A note on naturally embedded ternary trees, arXiv:0902.2646 [math.CO], 2009.
FORMULA
G.f.: (T(z) - 2)*T^3(z)/(T^2(z) - 3*T(z) + 1), where T(z) = 1 + z*T^3(z) is the generating function of ternary trees - see A001764.
From Peter Bala, Feb 06 2022: (Start)
a(n) = (2/(n+1))*binomial(3*n,n) + Sum_{k=0..n} (-1)^(k+1)*Fibonacci(k+1)* binomial(3*n,n-k)*(n*(11*k+5)-2*k(k+1))/(n*(2*n+k+1)) for n >= 1. See Kuba, Corollary 1, p. 6.
O.g.f.: A(x) = (1/x)*(B(x) - 2)/(B(x) - 1), where B(x) = Sum_{n >= 0} 2*(3*n)!/((2*n+1)!*((n+1)!))*x^n is the o.g.f. of A000139. (End)
MAPLE
a:= proc(n) option remember; `if`(n<3, 1+n*(n-1),
((1349*n^2-2738*n+953)*n*a(n-1) -(5567*n^3-20114*n^2
+22439*n-7320)*a(n-2)-(3*(3*n-4))*(19*n-11)*(3*n-5)
*a(n-3))/((2*(2*n-1))*(n+1)*(19*n-30)))
end:
seq(a(n), n=0..30); # Alois P. Heinz, Jul 03 2017
MATHEMATICA
a[n_] := a[n] = If[n < 3, 1 + n*(n - 1), ((1349*n^2 - 2738*n + 953)*n*a[n - 1] - (5567*n^3 - 20114*n^2 + 22439*n - 7320)*a[n - 2] - (3*(3*n - 4)) * (19*n - 11)*(3*n - 5)*a[n - 3])/((2*(2*n - 1))*(n + 1)*(19*n - 30))];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 09 2017, after Alois P. Heinz *)
PROG
(PARI) N=66; x='x+O('x^N); T=serreverse(x-x^3)/x; v=Vec(((T-2)*T^3/(T^2-3*T+1))); vector(#v\2, n, v[2*n-1]) \\ Joerg Arndt, May 26 2016
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Markus Kuba, Dec 08 2013
EXTENSIONS
More terms from F. Chapoton, May 26 2016
STATUS
approved