%I
%S 1,2,10,89,1156,19897,428002,11067457,334667368,11593751921,
%T 452892057454,19699549177585,944416040000044,49480473036710185,
%U 2812998429218735986,172475808692526176513,11345688093224067380176
%N Number of fat trees on n labeled vertices.
%C A fat tree on vertex set V is a partition of V together with edges (between vertices, not parts) that link the parts of the partition in a treelike pattern: that is, when the parts are collapsed to points, the edges are a (free) tree. A fat tree is in a (multi)graph G when the edges are edges of G. The fat forests in a graph form a geometric lattice.
%C If a(n) is the number of fat trees when each edge is replaced by M distinguishable copies of itself, then a(1) = 1, a(2) = M + 1, a(3) = 3 M^2 + 6 M + 1, a(4) = 16 M^3 + 48 M^2 + 24 M + 1, a(5) = 125 M^4 + 500 M^3 + 450 M^2 + 80 M + 1, a(6) = 1296 M^5 + 6480 M^4 + 8640 M^3 + 3240 M^2 + 240 M + 1.
%D Thomas Zaslavsky, "Perpendicular dissections of space". Discrete Comput. Geom., 27 (2002), 303351. MR 2003i:52026. Zbl. 1001.52011.
%H T. D. Noe, <a href="/A055779/b055779.txt">Table of n, a(n) for n=1..100</a>
%H Vaclav Kotesovec, <a href="http://members.chello.cz/kotesovec/math_articles/kotesovec_fat_tree_asymptotics.pdf">Asymptotic formula for number of fat trees on n labeled vertices</a>, Aug 25 2012, in Czech, main results in English.
%H T. Zaslavsky, <a href="https://arxiv.org/abs/1001.4435">Perpendicular dissections of space</a>, arXiv:1001.4435 [math.CO], 2010.
%F a(n) = Sum_{k=1..n} binomial(n, k)*k^(nk)*n^(k2).  _Vladeta Jovovic_, Jun 16 2006
%F a(n) = n!/n^2 sum_{mu a partition of n} product_j n^{mu_j}/(mu_j! (j1)!^{mu_j}), where mu_j is the number of parts of size j in the partition mu.  _Vladeta Jovovic_, Jun 15 2006
%F limit n>infinity (a(n)^(1/n))/n = (1p)^(p1)*p^(12*p) ~ 1.6554879129915343..., where p ~ 0.6924583254616546... is the root of the equation exp(11/p)=(1p)/p^2. [From _Vaclav Kotesovec_, Aug 25 2012]
%e For n=3, there is one fat tree with a single node, three with three nodes (choose which vertex to have in the middle) and six with two nodes (3 choices for which vertex to have by itself and 2 choices for which of the others to join it to).
%t Table[Sum[Binomial[n,k]k^(nk) n^(k2),{k,n}],{n,20}] (* _Harvey P. Dale_, Aug 24 2016 *)
%o (PARI) A055779(n) = sum(k=1, n, binomial(n, k)*k^(nk)*n^(k2)).  _Franklin T. AdamsWatters_, Jun 16 2006
%K nonn,nice,easy
%O 1,2
%A _Thomas Zaslavsky_, Jul 12 2000
%E Edited with more terms by _Franklin T. AdamsWatters_, Jun 13 2006
%E More terms from _Vladeta Jovovic_ and _Franklin T. AdamsWatters_, Jun 15 2006
