A108524 Number of ordered rooted trees with n generators.

1, 2, 7, 32, 166, 926, 5419, 32816, 203902, 1292612, 8327254, 54358280, 358769152, 2390130038, 16051344307, 108548774240, 738563388214, 5052324028508, 34727816264050, 239733805643552, 1661351898336676, 11553558997057772, 80603609263563262, 563972937201926432

Offset: 1

Author

Christian G. Bower, Jun 07 2005

Keywords

• nonn

Comments

A generator is a leaf or a node with just one child.

The Hankel transform of this sequence is 3^C(n+1,2). The Hankel transform of this sequence with 1 prepended (1,1,2,7,...) is 3^C(n,2). - Paul Barry, Jan 26 2011

a(n) is the number of Schroder paths of semilength n-1 in which the (2,0)-steps that are not on the horizontal axis come in 2 colors. Example: a(3)=7 because we have HH, UDUD, UUDD, HUD, UDH, UBD, and URD, where U=(1,1), H=(2,0), D=(1,-1), while B and R are, respectively, blue and red (2,0)-steps. - Emeric Deutsch, May 02 2011

Also the compositional inverse of A327767. - Peter Luschny, Oct 08 2022

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..200
• Paul Barry, Generalized Eulerian Triangles and Some Special Production Matrices, arXiv:1803.10297 [math.CO], 2018.
• Shishuo Fu, Yaling Wang, Bijective recurrences concerning two Schröder triangles, arXiv:1908.03912 [math.CO], 2019.
• Index entries for sequences related to rooted trees

Crossrefs

Cf. A108521-A108529, A108525, A000108, A001003, A327767.

Programs

Maple

# Using function CompInv from A357588.
CompInv(24, n -> [1, -2][irem(n-1, 2) + 1]); # Peter Luschny, Oct 08 2022

Mathematica

Rest[CoefficientList[Series[(Sqrt[4*x^2-8*x+1]-1)/(2*x-4), {x, 0, 20}], x]] (* Vaclav Kotesovec, Oct 18 2012 *)

Maxima

a(n):=sum((i*binomial(n+1,i)*sum((-1)^j*2^(n-j)*binomial(n,j)*binomial(2*n-j-i-1,n-1),j,0,n-i))/2^i,i,1,n+1)/(n*(n+1)); /* Vladimir Kruchinin, May 10 2011 */

Formula

G.f.: (sqrt(4*x^2-8*x+1) - 1)/(2*x-4).

G.f.: 1/(1-x-x/(1-2x-x/(1-2x-x/(1-2x-x/(1-2x-x/(1-... (continued fraction). - Paul Barry, Feb 10 2009

a(n) = sum(i=1..n+1, (i*C(n+1,i)*sum(j=0..n-i, (-1)^j*2^(n-j)*C(n,j)*C(2*n-j-i-1,n-1)))/2^i)/(n*(n+1)). - Vladimir Kruchinin, May 10 2011

From Gary W. Adamson, Jul 11 2011: (Start)

a(n) is upper left term in the following infinite square production matrix:

1, 1, 0, 0, 0, ...

1, 1, 1, 0, 0, ...

3, 3, 1, 1, 0, ...

9, 9, 3, 1, 1, ...

...

where columns are (1, 1, 3, 9, 27, 81, ...) prefaced with (0,0,1,2,3,...) zeros. (End)

Conjecture: 2*n*a(n) +(24-17*n)*a(n-1) +4*(4*n-9)*a(n-2) +4*(3-n)*a(n-3)=0. - R. J. Mathar, Nov 14 2011

G.f.: A(x)=(sqrt(4*x^2-8*x+1) - 1)/x/(2*x-4) = 1/(G(0)-x); G(k) = 1 + 2*x - 3*x/G(k+1); (continued fraction, 1-step ). - Sergei N. Gladkovskii, Jan 05 2012

a(n) ~ 3^(1/4)*(3^(3/2)-5)*(4+2*sqrt(3))^n/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 18 2012

From Peter Bala, Mar 13 2015: (Start)

The o.g.f. A(x) satisfies the differential equation (2 - 17*x + 16*x^2 - 4*x^3)A'(x) + (7 - 4*x)*A(x) = 2 - 2*x. Mathar's conjectural recurrence above follows from this.

The o.g.f. A(x) is the series reversion of the rational function x*(1 - 2*x)/(1 - x^2). (End)