A227917 Number of semi-increasing binary plane trees with n vertices.

1, 4, 26, 232, 2624, 35888, 575280, 10569984, 218911872, 5044346112, 127980834816, 3544627393536, 106408500206592, 3441351475359744, 119279906031888384, 4410902376303722496, 173335758665503997952, 7213199863532804702208, 316878056718379090771968

Offset

1,2

Comments

a(n) is the number of semi-increasing plane binary trees with n vertices, which are labeled binary plane trees where each vertex with two children has a label less than the label of each of its descendants.

Links

Brad R. Jones, Table of n, a(n) for n = 1..100

B. R. Jones, On tree hook length formulas, Feynman rules and B-series, Master's thesis, Simon Fraser University, 2014.

Formula

E.g.f.: 2/(2+log(1-2*x))-1.

E.g.f. A(x) satisfies the differential equation A'(x) = (1+2*A(x)+A(x)^2)/(1-2*x).

a(n) ~ n! * 2^(n+1)*exp(2*n)/(exp(2)-1)^(n+1). - Vaclav Kotesovec, Oct 30 2013

a(n) = Sum_{k=1..n} |Stirling1(n,k)| * k! * 2^(n-k). - Ilya Gutkovskiy, Apr 26 2021

Example

Examples of some semi-increasing binary plane trees of 4 vertices:
  ----------
      1
     / \
    4   2
   /
  3
  ----------
      1
     / \
    3   2
   /
  4
  ----------
      3
     /
    1
   / \
  4   2
  ----------
      3
     /
    1
     \
      2
       \
        4
  ----------
      1
     /
    2
     \
      3
     /
    4
  ----------
The following is NOT a semi-increasing binary tree because vertex 2 has two children and has vertex 1 as a descendant.
  ----------
      2
     / \
    3   4
   /
  1
  ----------

Maple

seq(coeff(taylor(2/(2+log(1-2*z))-1, z, 51), z^i)*i!, i=1..50);

Mathematica

Rest[CoefficientList[Series[2/(2+Log[1-2*x])-1, {x, 0, 20}], x]*Range[0, 20]!] (* Vaclav Kotesovec, Oct 30 2013 *)

Crossrefs

Sequence in context: A293915 A107879 A066224 * A136227 A346978 A000310

Adjacent sequences: A227914 A227915 A227916 * A227918 A227919 A227920

Keyword

nonn

Author

Brad R. Jones, Oct 22 2013

Status

approved