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
KEYWORD
nonn
AUTHOR
Brad R. Jones, Oct 22 2013
STATUS
approved