A052888 E.g.f. is series reversion of log(1+x)*exp(-x).

0, 1, 3, 19, 189, 2576, 44683, 941977, 23388025, 668520163, 21622993111, 780789908240, 31135480907413, 1358965445353621, 64440211018897379, 3298807094967155971, 181322497435007616497, 10651131815012588324380, 665881649529214120845679, 44144097851022253967955749

Offset: 0

Author

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Keywords

• easy
• nonn

Comments

A simple grammar.

For n > 0, Sum_{k=1..n} a(k)*Sum_{i=0..n-k} (-1)^i*k^i*Stirling1(n-i,k)/(i!*(n-i)!) = delta(n,1). - Vladimir Kruchinin, Feb 08 2012

From Gus Wiseman, Jul 20 2013: (Start)

Number of tail-trees of weight n. A tail is a pairing of a block of a set partition p with an element of some other block. A tail-tree on p is composed of a root block r, a tail-tree on each block of a set partition of the remaining blocks, and a tail from each of their roots to r.

On any set partition of weight n and length m, the total number of tail-forests with k components is equal to binomial(m-1, k-1)*n^(m-k). (End)

From Paul Laubie, Aug 25 2023: (Start)

Number of nonempty forests of rooted labeled hypertrees with a total number of vertices equal to n.

E.g., n=1: Only one forest is possible, which is {1}, the forest with one hypertree with one vertex.

n=2: Three forests are possible: {1,2}, the forest with two hypertrees, each having one vertex labeled 1 for one on the hypertree and 2 for the other hypertree; the forest {1-2}, with only one hypertree, with two vertices rooted at 1; and the forest {2-1}, with only one hypertree, with two vertices rooted at 2. (End)

Links

• G. C. Greubel, Table of n, a(n) for n = 0..368
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 863
• Rosena R. X. Du and Fu Liu, Pure-cycle Hurwitz factorizations and multi-noded rooted trees, arXiv:1008.3677 [math.CO], 2010-2013.
• Gus Wiseman, All 189 tail-trees of weight 4.
• Gus Wiseman, Set maps, umbral calculus, and the chromatic polynomial, Discrete Math., 308(16):3551-3564, 2008.

Programs

Maple

spec := [S,{C=Prod(Z,B),B=Set(S),S=Set(C,1 <= card)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);

Mathematica

Table[Sum[StirlingS2[n,k]*n^(k-1),{k,1,n}],{n,0,20}] (* Vaclav Kotesovec, Jan 22 2014 *)

PARI

for(n=0,30, print1(sum(k=1,n, stirling(n,k,2)*n^(k-1)), ", ")) \\ G. C. Greubel, Nov 17 2017

Formula

E.g.f.: RootOf(_Z-exp(exp(_Z)*x)+1)

a(n) = Sum_{k=1..n} Stirling2(n, k)*n^(k-1). - Vladeta Jovovic, Jul 26 2005

a(n) = exp(-n)*Sum_{k>=1} n^(k-1)*k^n/k!. - Vladeta Jovovic, Jul 03 2006 [corrected by Ilya Gutkovskiy, Apr 20 2020]

a(n) ~ exp(n*(LambertW(1) + 1/LambertW(1) - 2)) * n^(n-1) / sqrt(1+LambertW(1)). - Vaclav Kotesovec, Jan 22 2014