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A052888
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E.g.f. is series reversion of log(1+x)*exp(-x).
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32
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0, 1, 3, 19, 189, 2576, 44683, 941977, 23388025, 668520163, 21622993111, 780789908240, 31135480907413, 1358965445353621, 64440211018897379, 3298807094967155971, 181322497435007616497, 10651131815012588324380, 665881649529214120845679, 44144097851022253967955749
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OFFSET
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0,3
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COMMENTS
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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
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)
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)
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LINKS
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FORMULA
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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*(LambertW(1) + 1/LambertW(1) - 2)) * n^(n-1) / sqrt(1+LambertW(1)). - Vaclav Kotesovec, Jan 22 2014
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MAPLE
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spec := [S, {C=Prod(Z, B), B=Set(S), S=Set(C, 1 <= card)}, labeled]: seq(combstruct[count](spec, size=n), n=0..20);
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MATHEMATICA
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Table[Sum[StirlingS2[n, k]*n^(k-1), {k, 1, n}], {n, 0, 20}] (* Vaclav Kotesovec, Jan 22 2014 *)
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PROG
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(PARI) for(n=0, 30, print1(sum(k=1, n, stirling(n, k, 2)*n^(k-1)), ", ")) \\ G. C. Greubel, Nov 17 2017
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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encyclopedia(AT)pommard.inria.fr, Jan 25 2000
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STATUS
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approved
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