%I A055779
%S A055779 1,2,10,89,1156,19897,428002,11067457,334667368,11593751921,
%T A055779 452892057454,19699549177585,944416040000044,49480473036710185,
%U A055779 2812998429218735986,172475808692526176513,11345688093224067380176
%N A055779 Number of fat trees on n labeled vertices.
%C A055779 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 tree-like 
               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 A055779 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 A055779 Thomas Zaslavsky, "Perpendicular dissections of space". Discrete Comput. 
               Geom., 27 (2002), 303-351. MR 2003i:52026. Zbl. 1001.52011.
%H A055779 T. D. Noe, <a href="b055779.txt">Table of n, a(n) for n=1..100</a>
%F A055779 a(n) = Sum_{k=1..n} binomial(n, k)*k^(n-k)*n^(k-2). - Vladeta Jovovic 
               (vladeta(AT)eunet.rs), Jun 16 2006
%F A055779 a(n) = n!/n^2 sum_{mu a partition of n} product_j n^{mu_j}/(mu_j! (j-1)!^{mu_j}), 
               where mu_j is the number of parts of size j in the partition mu. 
               - Vladeta Jovovic (vladeta(AT)eunet.rs), Jun 15 2006
%e A055779 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).
%o A055779 (PARI) A055779(n) = local(k);sum(k=1, n, binomial(n, k)*k^(n-k)*n^(k-2)). 
               - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Jun 16 2006
%Y A055779 Sequence in context: A144002 A060350 A096658 this_sequence A067550 A086587 
               A082472
%Y A055779 Adjacent sequences: A055776 A055777 A055778 this_sequence A055780 A055781 
               A055782
%K A055779 nonn,nice,easy
%O A055779 1,2
%A A055779 Thomas Zaslavsky (zaslav(AT)math.binghamton.edu), Jul 12 2000
%E A055779 Edited with more terms by Frank Adams-Watters (FrankTAW(AT)Netscape.net), 
               Jun 13 2006
%E A055779 More terms from Vladeta Jovovic (vladeta(AT)eunet.rs) and Frank Adams-Watters 
               (FrankTAW(AT)Netscape.net), Jun 15 2006