cp's OEIS Frontend

A006241 Number of minimal plane trees with n terminal nodes.

%I A006241 M0133 #35 Dec 26 2021 21:31:59
%S A006241 1,1,1,2,1,3,1,6,2,3,1,20,1,3,3,54,1,34,1,44,3,3,1,764,2,3,10,140,1,
%T A006241 283,1,4470,3,3,3,10416,1,3,3,10820,1,2227,1,2060,62,3,1,958476,2,250,
%U A006241 3,8204,1,59154,3,316004,3,3,1,3457904,1,3,158,30229110,3
%N A006241 Number of minimal plane trees with n terminal nodes.
%C A006241 In equation (4.4) Lew says a(p^3) = 3+3^p, but this is incorrect, it should be a(p^3) = 2+2^p. - _Sean A. Irvine_, Feb 07 2017
%C A006241 From _Gus Wiseman_, Jan 15 2017: (Start)
%C A006241 Number of same-trees of weight n with all leaves equal to 1. A same-tree is either: (case 1) a positive integer, or (case 2) a finite sequence of two or more same-trees all having the same weight, where the weight in case 2 is the sum of weights.
%C A006241 For n>1, a(n) is also equal to the number of same-trees of weight n with all leaves greater than 1 (see example). (End)
%D A006241 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006241 Vincenzo Librandi, <a href="/A006241/b006241.txt">Table of n, a(n) for n = 1..1000</a>
%H A006241 J. S. Lew, <a href="http://dx.doi.org/10.1007/BF01776577">Polynomial enumeration of multidimensional lattices</a>, Math. Systems Theory, 12 (1978), 253-270.
%H A006241 Gus Wiseman, <a href="/A006241/a006241.png">Same trees with all leaves equal to one n=1..15</a>
%H A006241 <a href="/index/Tra#trees">Index entries for sequences related to trees</a>
%F A006241 a(1)=a(2)=a(3)=a(5)=a(7)=1, a(4)=2, a(6)=3, a(n) = Sum_{1 != d | n} a(n / d)^d [From Lew]. - _Sean A. Irvine_, Feb 07 2017 [typo corrected by _Ilya Gutkovskiy_, Apr 24 2019]
%e A006241 The a(12)=20 same-trees with all leaves greater than 1 are:
%e A006241 12, (3333), (222222), ((33)(33)), ((33)(222)), ((33)6), ((222)(33)), ((222)(222)), ((222)6), (6(33)), (6(222)), (66), ((22)(22)(22)), ((22)(22)4), ((22)4(22)), ((22)44), (4(22)(22)), (4(22)4), (44(22)), (444). - _Gus Wiseman_, Jan 15 2017
%p A006241 a:= proc(n) option remember; `if`(n=1, 1, add(
%p A006241       a(n/d)^d, d=numtheory[divisors](n) minus {1}))
%p A006241     end:
%p A006241 seq(a(n), n=1..70);  # _Alois P. Heinz_, Feb 21 2017
%t A006241 Array[If[#1===1,1,Sum[#0[#1/d]^d,{d,Rest[Divisors[#1]]}]]&,200] (* _Gus Wiseman_, Jan 15 2017 *)
%Y A006241 Cf. A001003, A281145.
%K A006241 nonn
%O A006241 1,4
%A A006241 _N. J. A. Sloane_
%E A006241 a(8), a(27), and a(50) corrected by _Sean A. Irvine_, Feb 07 2017