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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.

%I A005174 M4738 #50 Mar 28 2025 16:07:10
%S A005174 0,0,10,124,890,5060,25410,118524,527530,2276020,9613010,40001324,
%T A005174 164698170,672961380,2734531810,11066546524,44652164810,179768037140,
%U A005174 722553165810,2900661482124,11634003919450,46630112719300,186802788139010,748058256616124
%N 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.
%D A005174 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005174 Andrew Howroyd, <a href="/A005174/b005174.txt">Table of n, a(n) for n = 1..500</a>
%H A005174 F. R. McMorris and T. Zaslavsky, <a href="http://dx.doi.org/10.1016/0025-5564(81)90071-7">The number of cladistic characters</a>, Math. Biosciences, 54 (1981), 3-10.
%H A005174 F. R. McMorris and T. Zaslavsky, <a href="/A005172/a005172.pdf">The number of cladistic characters</a>, Math. Biosciences, 54 (1981), 3-10. [Annotated scanned copy]
%H A005174 Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de séries génératrices et quelques conjectures</a>, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
%H A005174 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H A005174 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%H A005174 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (10,-35,50,-24).
%F A005174 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
%F A005174 Formula of Layman matches the proven formula in McMorris and Zaslavsky. - _Sean A. Irvine_, Apr 12 2016
%F A005174 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
%F A005174 G.f.: 2*x^3*(5 + 12*x)/((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)). - _Andrew Howroyd_, Mar 28 2025
%p A005174 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
%Y A005174 Column 4 of A094262.
%K A005174 nonn,easy
%O A005174 1,3
%A A005174 _N. J. A. Sloane_
%E A005174 Name clarified by _Andrew Howroyd_, Mar 28 2025