cp's OEIS Frontend

A318227 Number of inequivalent leaf-colorings of rooted identity trees with n nodes.

%I A318227 #11 Dec 14 2020 01:36:51
%S A318227 1,1,1,3,5,14,38,114,330,1054,3483,11841,41543,149520,552356,2084896,
%T A318227 8046146,31649992,127031001,518434863,2153133594,9081863859,
%U A318227 38909868272,169096646271,745348155211,3329032020048,15063018195100,68998386313333,319872246921326,1500013368166112
%N A318227 Number of inequivalent leaf-colorings of rooted identity trees with n nodes.
%C A318227 In a rooted identity tree, all branches directly under any given branch are different.
%C A318227 The leaves are colored after selection of the tree. Since all trees are asymmetric, the symmetry group of the leaves will be the identity group and a tree with k leaves will have Bell(k) inequivalent leaf-colorings. - _Andrew Howroyd_, Dec 10 2020
%H A318227 Andrew Howroyd, <a href="/A318227/b318227.txt">Table of n, a(n) for n = 1..200</a>
%F A318227 a(n) = Sum_{k=1..n} A055327(n,k) * A000110(k). - _Andrew Howroyd_, Dec 10 2020
%e A318227 Inequivalent representatives of the a(6) = 14 leaf-colorings:
%e A318227   (1(1(1)))  ((1)((1)))  (1(((1))))  ((1((1))))  (((1(1))))  (((((1)))))
%e A318227   (1(1(2)))  ((1)((2)))  (1(((2))))  ((1((2))))  (((1(2))))
%e A318227   (1(2(1)))
%e A318227   (1(2(2)))
%e A318227   (1(2(3)))
%t A318227 idt[n_]:=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[idt/@c]],UnsameQ@@#&],{c,IntegerPartitions[n-1]}]];
%t A318227 Table[Sum[BellB[Count[tree,{},{0,Infinity}]],{tree,idt[n]}],{n,16}]
%o A318227 (PARI) \\ bell(n) is A000110(n).
%o A318227 WeighMT(u)={my(n=#u, p=x*Ser(u), vars=variables(p)); Vec(exp( sum(i=1, n, (-1)^(i-1)*substvec(p + O(x*x^(n\i)), vars, apply(v->v^i,vars))/i ))-1)}
%o A318227 bell(n)={sum(k=1, n, stirling(n,k,2))}
%o A318227 seq(n)={my(v=[y], b=vector(n,k,bell(k))); for(n=2, n, v=concat(v[1], WeighMT(v))); vector(n, k, sum(i=1, k, polcoef(v[k],i)*b[i]))} \\ _Andrew Howroyd_, Dec 10 2020
%Y A318227 Cf. A000081, A001190, A001678, A003238, A004111, A290689, A318185, A304486.
%Y A318227 Cf. A318226, A318228, A318229, A318230, A318231, A318234.
%Y A318227 Cf. A000110 (Bell numbers), A055327, A301342.
%K A318227 nonn
%O A318227 1,4
%A A318227 _Gus Wiseman_, Aug 21 2018
%E A318227 Terms a(17) and beyond from _Andrew Howroyd_, Dec 10 2020