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
Keywords
Examples
Examples of some semi-increasing binary plane trees of 4 vertices:
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1
/ \
4 2
/
3
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1
/ \
3 2
/
4
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3
/
1
/ \
4 2
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3
/
1
\
2
\
4
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1
/
2
\
3
/
4
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The following is NOT a semi-increasing binary tree because vertex 2 has two children and has vertex 1 as a descendant.
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2
/ \
3 4
/
1
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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.
Programs
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Maple
seq(coeff(taylor(2/(2+log(1-2*z))-1, z, 51), z^i)*i!, i=1..50);
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Mathematica
Rest[CoefficientList[Series[2/(2+Log[1-2*x])-1, {x,0,20}], x]*Range[0,20]!] (* Vaclav Kotesovec, Oct 30 2013 *)
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
Comments