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A052888 E.g.f. is series reversion of log(1+x)*exp(-x). 19
0, 1, 3, 19, 189, 2576, 44683, 941977, 23388025, 668520163, 21622993111, 780789908240, 31135480907413, 1358965445353621, 64440211018897379, 3298807094967155971, 181322497435007616497, 10651131815012588324380 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

A simple grammar.

For n>0 is true sum(a(k)*sum(((-1)^i*k^i*stirling1(n-i,k))/(i!*(n-i)!),i,0,n-k),k,1,n)=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)

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, Fu Liu, Pure-cycle Hurwitz factorizations and multi-noded rooted trees, arXiv:1008.3677 [math.CO]

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.

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

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

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 *)

PROG

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

CROSSREFS

Sequence in context: A202617 A143633 A326553 * A141623 A229234 A090354

Adjacent sequences:  A052885 A052886 A052887 * A052889 A052890 A052891

KEYWORD

easy,nonn

AUTHOR

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

STATUS

approved

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Last modified January 26 20:33 EST 2020. Contains 331288 sequences. (Running on oeis4.)