A011270 Hybrid binary rooted trees with n nodes whose root is labeled by "n".

1, 1, 4, 18, 90, 481, 2690, 15547, 92124, 556664, 3417062, 21248966, 133576724, 847465593, 5419399722, 34895368578, 226050057378, 1472170887755, 9633297762870, 63305402213336, 417612181048826, 2764492667188504, 18358282050480384, 122265756020847943

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.

Index entries for reversions of series

Index entries for sequences related to rooted trees

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.

Sequence in context: A036749 A218760 A219305 * A367724 A355247 A269450

Adjacent sequences: A011267 A011268 A011269 * A011271 A011272 A011273

Keyword

nonn

Author

pallo(AT)u-bourgogne.fr (Jean Pallo)

Status

approved