OFFSET
0,3
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..500
Nancy S. S. Gu, Nelson Y. Li, and Toufik Mansour, 2-Binary trees: bijections and related issues, Discr. Math., 308 (2008), 1209-1221.
J. M. Pallo, On the listing and random generation of hybrid binary trees, International Journal of Computer Mathematics, 50, 1994, 135-145.
FORMULA
G.f.: = 1+x*G(x)^2, where G(x) is g.f. for A007863.
Reversion of x - (x/(1 - x))^2 = 0, 1, -1, -2, -3, -4, -5, ... - Olivier Gérard, Jul 05 2001
a(n) = (2/(n+2))*Sum_{j=0...n} binomial(n+j+1, n+1)*binomial(n+j+2, n-j). - Vladimir Kruchinin, Dec 24 2010
G.f. A(x) satisfies: A(x) = 1/(1 - Sum_{k>=1} k*x^k*A(x)^k). - Ilya Gutkovskiy, Apr 10 2018
G.f. A(x) satisfies: A(x) = 1 + Sum_{n>=1} n^(n-1) * x^n*A(x)^(n+1) / (1 + (n-1)*x*A(x))^(n+1). - Paul D. Hanna, Oct 08 2023
a(n) ~ sqrt((35 + (869750 - 5250*sqrt(105))^(1/3) + 5*(14*(497 + 3*sqrt(105)))^(1/3))/525) / (sqrt(Pi) * n^(3/2) * ((2 - 104/(-181 + 105*sqrt(105))^(1/3) + (-181 + 105*sqrt(105))^(1/3))/6)^n). - Vaclav Kotesovec, Oct 08 2023
EXAMPLE
G.f. A(x) = 1 + x + 4*x^2 + 18*x^3 + 90*x^4 + 481*x^5 + 2690*x^6 + 15547*x^7 + 92124*x^8 + 556664*x^9 + 3417062*x^10 + ...
where x = x*A(x) - x^2*A(x)^2/(1 - x*A(x))^2.
MAPLE
G:= proc(n) option remember; if n<=0 then 1 else convert(series(
(x^2*G(n-1)^3 +x*G(n-1)^2 +1)/ (1-x), x=0, n+1), polynom) fi
end:
a:= n-> coeff(1+x*G(n-1)^2, x, n):
seq(a(n), n=0..20); # Alois P. Heinz, Aug 22 2008
# second Maple program:
a:= proc(n) option remember; `if`(n<3, [1, 1, 4][n+1], (
6*n*(210*n^2-411*n+163)*a(n-1)-4*(n-2)*(7*n-6)*(5*n-3)*a(n-2)
+2*(n-3)*(2*n-3)*(35*n-16)*a(n-3))/(5*n*(n+1)*(35*n-51)))
end:
seq(a(n), n=0..25); # Alois P. Heinz, May 18 2013
MATHEMATICA
a[0] = 1; a[n_] := n*HypergeometricPFQ[{1-n, n+1, n+2}, {3/2, 2}, -1/4]; Table[ a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 02 2015, after Vladimir Kruchinin *)
CROSSREFS
Cf. A011272.
KEYWORD
nonn
AUTHOR
pallo(AT)u-bourgogne.fr (Jean Pallo)
STATUS
approved