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URL: https://oeis.org/A011270

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A011270
Hybrid binary rooted trees with n nodes whose root is labeled by "n".
11
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
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
# Alternative:
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
Sequence in context: A036749 A218760 A219305 * A367724 A355247 A269450
KEYWORD
nonn
AUTHOR
Jean Pallo (pallo(AT)u-bourgogne.fr)
STATUS
approved