OFFSET
0,3
COMMENTS
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
a(n) is the number of distinct labeled increasing trees of size n (see Wirtz at pages 4 - 5). - Stefano Spezia, Nov 13 2022
a(n) is the number of labeled histories possible across all at-most-(n+1)-furcating trees with n+1 leaves. - Noah A Rosenberg, May 13 2026
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..260
Christian Blatter, How many ways to merge N companies into one big company: Bell or Catalan?, Math StackExchange.
Emily H. Dickey and Noah A. Rosenberg, Labeled histories and maximally probable labeled topologies with multifurcation, arXiv:2511.16799 [q-bio.PE], 2025. See pp. 4, 12.
Emily H. Dickey and Noah A. Rosenberg, Labeled histories and maximally probable labeled topologies with multifurcation, Discr. Appl. Math. 391 (2026), 192-203. See Table 1.
Johannes Wirtz, On the enumeration of leaf-labeled increasing trees with arbitrary node-degree, arXiv:2211.03632 [q-bio.PE], 2022.
Johannes Wirtz, On the enumeration of leaf-labeled increasing trees with arbitrary node-degree, Art Discr. Appl. Math. 7 (2024), #P1.10. See Table 1.
FORMULA
a(n) ~ c * n^(2*n+8/3) / (2^n * exp(2*n)), where c = 4.001655169623968944922713533374039000521095549333460838578... .
E.g.f. A(x) satisfies A(x) = A'(x)*(exp(x) - 1 - x) + x (see Wirtz 2022 at page 7). - Stefano Spezia, Nov 13 2022
MATHEMATICA
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, May 06 2015
STATUS
approved