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%I #22 Nov 13 2022 10:36:12
%S 1,1,4,29,336,5687,132294,4047969,157601068,7607093435,445794008034,
%T 31177310522789,2564976392355144,245223349515360543,
%U 26959450820298057694,3377267272710103354409,478240674001176206987556,76011318838172580152245187
%N Recurrence: a(n) = Sum_{k=0..n-1} a(k)*C(n+1,k), a(0)=1.
%C The number of merger histories for n+1 distinct firms into one firm, where any number of firms may be merged at a time but mergers of two separate sets of firms never occur simultaneously. See MathStackExchange link. - _William P. Orrick_, Oct 28 2016
%C a(n) is the number of distinct labeled increasing trees of size n (see Wirtz at pages 4 - 5). - _Stefano Spezia_, Nov 13 2022
%H Vaclav Kotesovec, <a href="/A256006/b256006.txt">Table of n, a(n) for n = 0..260</a>
%H Christian Blatter, <a href="http://math.stackexchange.com/a/214203/3736">How many ways to merge N companies into one big company: Bell or Catalan?</a>, Math StackExchange.
%H Johannes Wirtz, <a href="https://arxiv.org/abs/2211.03632">On the enumeration of leaf-labelled increasing trees with arbitrary node-degree</a>, arXiv:2211.03632 [q-bio.PE], 2022.
%F a(n) ~ c * n^(2*n+8/3) / (2^n * exp(2*n)), where c = 4.001655169623968944922713533374039000521095549333460838578... .
%F E.g.f. A(x) satisfies A(x) = A'(x)*(exp(x) - 1 - x) + x (see Wirtz at page 7). - _Stefano Spezia_, Nov 13 2022
%t nmax = 30; aa = ConstantArray[0,nmax+1]; aa[[1]] = 1; Do[aa[[n+1]]=Sum[Binomial[n+1,k]*aa[[k+1]],{k,0,n-1}],{n,nmax}]; aa
%Y Cf. A103996.
%K nonn
%O 0,3
%A _Vaclav Kotesovec_, May 06 2015
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