A011272 -id:A011272 - cp's OEIS frontend

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

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

nonn

			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.

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

Cf. A011272.

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

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 *)

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

A011274 Triangle of numbers of hybrid rooted trees (divided by Fibonacci numbers).

Original entry on oeis.org

1, 2, 1, 7, 4, 1, 31, 18, 6, 1, 154, 90, 33, 8, 1, 820, 481, 185, 52, 10, 1, 4575, 2690, 1065, 324, 75, 12, 1, 26398, 15547, 6276, 2006, 515, 102, 14, 1, 156233, 92124, 37711, 12468, 3420, 766, 133, 16, 1, 943174, 556664, 230277, 78030, 22412, 5439, 1085, 168, 18, 1

Offset: 1

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

Triangle T(n,k) = [x^(n-k)] A(x)^k where A(x) is the o.g.f. of A007863. - Vladimir Kruchinin, Mar 17 2011

Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A007863. - Philippe Deléham, Feb 03 2014

			     1
     2    1
     7    4    1
    31   18    6   1
   154   90   33   8  1
   820  481  185  52 10  1
  4575 2690 1065 324 75 12 1
Production matrix is:
   2   1
   3   2   1
   5   3   2   1
   8   5   3   2   1
  13   8   5   3   2   1
  21  13   8   5   3   2   1
  34  21  13   8   5   3   2   1
  55  34  21  13   8   5   3   2   1
  89  55  34  21  13   8   5   3   2   1
  ... - _Philippe Deléham_, Feb 03 2014

Vincenzo Librandi, Rows n = 1..100, flattened
Vladimir Kruchinin, D. V. Kruchinin, Composita and their properties, arXiv:1103.2582 [math.CO], 2011-2013.
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 sequences related to rooted trees

Cf. A000045, A011270, A011272.

A011274 := proc(n,k) k/n*add( binomial(i+n-1,n-1)*binomial(i+n,n-k-i),i=0..n-k) ; end proc: # R. J. Mathar, Mar 21 2011

t[n_, k_] := k/n*Binomial[n, k]*HypergeometricPFQ[ {k-n, n, n+1}, {1/2 + k/2, 1+k/2}, -1/4]; Flatten[ Table[ t[n, k], {n, 1, 10}, {k, 1, n}]] (* Jean-François Alcover, Dec 02 2011, after Vladimir Kruchinin *)

A011274(n,k):= k/n*sum(binomial(i+n-1,n-1)*binomial(i+n,n-k-i), i,0,n-k); /* Vladimir Kruchinin, Mar 17 2011 */

T(n,k) = (k/n) *Sum_{i=0..n-k} binomial(i+n-1,n-1)*binomial(i+n,n-k-i). - Vladimir Kruchinin, Mar 17 2011
(r/(m*n+r))*T((m+1)*n+r,m*n+r) = Sum_{k=1..n} k*T((m+1)*n-k,m*n)*T(r+k,r)/n. - Vladimir Kruchinin, Mar 17 2011
T(n,m) = (m/n)*Sum_{k=1..n-m+1} k*A007863(k-1)*T(n-k,m-1), 1 < m <= n. - Vladimir Kruchinin, Mar 17 2011

cp's OEIS Frontend

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

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