A005174 Number of rooted trees with 4 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, 0, 10, 124, 890, 5060, 25410, 118524, 527530, 2276020, 9613010, 40001324, 164698170, 672961380, 2734531810, 11066546524, 44652164810, 179768037140, 722553165810, 2900661482124, 11634003919450, 46630112719300, 186802788139010, 748058256616124
Offset: 1
References
- 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.
- 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 (10,-35,50,-24).
Crossrefs
Column 4 of A094262.
Programs
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Maple
A005174:=2*z**2*(5+12*z)/(z-1)/(3*z-1)/(2*z-1)/(4*z-1); # conjectured by Simon Plouffe in his 1992 dissertation
Formula
The terms a(1)-a(18) are given by a(n) = (8/3)*(4^n - 4) - 9*3^n + 11*2^n + 5. - John W. Layman, Jul 20 1999
Formula of Layman matches the proven formula in McMorris and Zaslavsky. - Sean A. Irvine, Apr 12 2016
E.g.f.: (1/3)*(-17*exp(x) + 66*exp(2*x) - 81*exp(3*x) + 32*exp(4*x)). - Ilya Gutkovskiy, Apr 12 2016
G.f.: 2*x^3*(5 + 12*x)/((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)). - Andrew Howroyd, Mar 28 2025
Extensions
Name clarified by Andrew Howroyd, Mar 28 2025