cp's OEIS Frontend

A005175 Number of rooted trees with 5 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 A005175 M3173 #49 Mar 28 2025 16:07:37
%S A005175 0,0,3,131,1830,16990,127953,851361,5231460,30459980,170761503,
%T A005175 931484191,4979773890,26223530970,136522672653,704553794621,
%U A005175 3611494269120,18415268221960,93516225653403,473366777478651,2390054857197150,12043393363764950,60590148885015753
%N A005175 Number of rooted trees with 5 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 A005175 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A005175 Andrew Howroyd, <a href="/A005175/b005175.txt">Table of n, a(n) for n = 1..500</a>
%H A005175 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 A005175 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 A005175 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 A005175 Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.
%H A005175 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%H A005175 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (15,-85,225,-274,120).
%F A005175 a(n+1) = 3*(3^n - 2*2^n + 1)/2 + 113*(4^n - 3*3^n + 3*2^n - 1)/6 + 625*(5^n - 4*4^n + 6*3^n - 4*2^n + 1)/24. - formula fitted by _John W. Layman_
%F A005175 a(n) = (125/24) * 5^n - (64/3) * 4^n + (135/4)*3^n - (76/3) * 2^n + 209/24 proven in McMorris and Zaslavsky, matches Layman's formula with an offset of 1. - _Sean A. Irvine_, Apr 12 2016
%F A005175 E.g.f.: (1/24)*exp(x)*(-1 + exp(x))^2*(209 - 798*exp(x) + 625*exp(2*x)). - _Ilya Gutkovskiy_, Apr 12 2016
%F A005175 G.f.: x^3*(3 + 86*x + 120*x^2)/((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 5*x)). - _Andrew Howroyd_, Mar 28 2025
%p A005175 A005175:=-z**2*(3+86*z+120*z**2)/(z-1)/(4*z-1)/(3*z-1)/(2*z-1)/(5*z-1); # conjectured by _Simon Plouffe_ in his 1992 dissertation
%t A005175 Table[(125/24) 5^n - (64/3) 4^n + (135/4) 3^n - (76/3) 2^n + 209/24, {n, 20}] (* _Michael De Vlieger_, Apr 12 2016 *)
%Y A005175 Column 5 of A094262.
%K A005175 nonn,easy
%O A005175 1,3
%A A005175 _N. J. A. Sloane_
%E A005175 Name clarified by _Andrew Howroyd_, Mar 28 2025