A005173 Number of rooted trees with 3 nodes of disjoint sets of labels with union {1..n}. If a node has an empty set of labels then it must have at least two children.
0, 1, 12, 61, 240, 841, 2772, 8821, 27480, 84481, 257532, 780781, 2358720, 7108921, 21392292, 64307941, 193185960, 580082161, 1741295052, 5225982301, 15682141200, 47054812201, 141181213812, 423577195861, 1270798696440, 3812530307041, 11437859356572, 34314114940621
Offset: 1
Examples
From _Andrew Howroyd_, Mar 28 2025: (Start)
The a(3) = 12 trees up to relabeling have one of the following 3 forms:
{} {1} {1}
/ \ / \ |
{1} {2,3} {2} {3} {2}
|
{3}
(End)
References
- F. R. McMorris and T. Zaslavsky, The number of cladistic characters, Math. Biosciences, 54 (1981), 3-10.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..500
- F. R. McMorris and T. Zaslavsky, The number of cladistic characters, Math. Biosciences, 54 (1981), 3-10. [Annotated scanned copy]
- Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
- Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992.
- Index entries for sequences related to trees
- Index entries for linear recurrences with constant coefficients, signature (6, -11, 6).
Programs
-
Maple
A005173:=-z*(1+6*z)/(z-1)/(3*z-1)/(2*z-1); # conjectured by Simon Plouffe in his 1992 dissertation
-
Mathematica
CoefficientList[Series[x (1+6 x)/(1-x)/(1-2 x)/(1-3 x),{x,0,30}],x] (* Harvey P. Dale, Jul 03 2023 *)
Formula
G.f.: x*(1 + 6*x) / ((1 - x)*(1 - 2*x)*(1 - 3*x)). [corrected by Ray Chandler, Jun 26 2023]
First differences give A003063, 3^(n-1) - 2^n.
From Andrew Howroyd, Mar 28 2025: (Start)
a(n) = (3^(n+1) - 2^(n+3) + 7)/2.
E.g.f.: (3*exp(x)/2 - 1)*(exp(x) - 1)^2. (End)
Extensions
More terms from Larry Reeves (larryr(AT)acm.org), Feb 06 2001
Name clarified by Andrew Howroyd, Mar 28 2025