A038049 Number of labeled rooted trees with 2-colored leaves.
2, 4, 24, 224, 2880, 47232, 942592, 22171648, 600698880, 18422374400, 630897721344, 23864653578240, 988197253808128, 44460603225407488, 2159714024218951680, 112652924603290615808, 6280048587936003784704, 372616014329572403183616, 23445082059018189741752320
Offset: 1
References
- F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 185 (3.1.83)
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..150
- Alexander Burstein and Louis W. Shapiro, Pseudo-involutions in the Riordan group, arXiv:2112.11595 [math.CO], 2021.
- N. J. A. Sloane, Transforms
- Index entries for sequences related to rooted trees
Programs
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Maple
a:= n-> add(binomial(n, k)*(n-k)^(n-1), k=0..n): seq(a(n), n=1..20); # Alois P. Heinz, Nov 30 2012
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Mathematica
Table[n!*Sum[2^j/j!*StirlingS2[n-1,n-j],{j,1,n}],{n,1,20}] (* Vaclav Kotesovec, Nov 30 2012 *)
Formula
Divides by n and shifts left under exponential transform.
E.g.f.: A(x) = x-LambertW(-x*exp(x)). - Vladeta Jovovic, Mar 08 2003
a(n) = Sum_{k=0..n} (binomial(n, k)*(n-k)^(n-1)).
A(x) = 2*compositional inverse of 2*x/(1+exp(2*x)). - Peter Bala, Oct 14 2011
a(n) ~ n^(n-1) * sqrt((1+LambertW(1/e))) / (e*LambertW(1/e))^n. - Vaclav Kotesovec, Nov 30 2012