A318227 - OEIS

%I #11 Dec 14 2020 01:36:51

%S 1,1,1,3,5,14,38,114,330,1054,3483,11841,41543,149520,552356,2084896,

%T 8046146,31649992,127031001,518434863,2153133594,9081863859,

%U 38909868272,169096646271,745348155211,3329032020048,15063018195100,68998386313333,319872246921326,1500013368166112

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

%C In a rooted identity tree, all branches directly under any given branch are different.

%C 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 Andrew Howroyd, <a href="/A318227/b318227.txt">Table of n, a(n) for n = 1..200</a>

%F a(n) = Sum_{k=1..n} A055327(n,k) * A000110(k). - _Andrew Howroyd_, Dec 10 2020

%e Inequivalent representatives of the a(6) = 14 leaf-colorings:

%e   (1(1(1)))  ((1)((1)))  (1(((1))))  ((1((1))))  (((1(1))))  (((((1)))))

%e   (1(1(2)))  ((1)((2)))  (1(((2))))  ((1((2))))  (((1(2))))

%e   (1(2(1)))

%e   (1(2(2)))

%e   (1(2(3)))

%t idt[n_]:=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[idt/@c]],UnsameQ@@#&],{c,IntegerPartitions[n-1]}]];

%t Table[Sum[BellB[Count[tree,{},{0,Infinity}]],{tree,idt[n]}],{n,16}]

%o (PARI) \\ bell(n) is A000110(n).

%o 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 bell(n)={sum(k=1, n, stirling(n,k,2))}

%o 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 Cf. A000081, A001190, A001678, A003238, A004111, A290689, A318185, A304486.

%Y Cf. A318226, A318228, A318229, A318230, A318231, A318234.

%Y Cf. A000110 (Bell numbers), A055327, A301342.

%K nonn

%O 1,4

%A _Gus Wiseman_, Aug 21 2018

%E Terms a(17) and beyond from _Andrew Howroyd_, Dec 10 2020