cp's OEIS Frontend

A330469 Number of series-reduced rooted trees whose leaves are multisets with a total of n elements covering an initial interval of positive integers.

%I A330469 #12 Feb 28 2020 12:56:23
%S A330469 1,1,4,24,250,3744,73408,1768088,50468854,1664844040,62304622320,
%T A330469 2607765903568,120696071556230,6120415124163512,337440974546042416,
%U A330469 20096905939846645064,1285779618228281270718,87947859243850506008984,6404472598196204610148232
%N A330469 Number of series-reduced rooted trees whose leaves are multisets with a total of n elements covering an initial interval of positive integers.
%C A330469 Also the number of different colorings of phylogenetic trees with n labels using a multiset of colors covering an initial interval of positive integers. A phylogenetic tree is a series-reduced rooted tree whose leaves are (usually disjoint) sets.
%H A330469 Andrew Howroyd, <a href="/A330469/b330469.txt">Table of n, a(n) for n = 0..200</a>
%e A330469 The a(3) = 24 trees:
%e A330469   (123)          (122)          (112)          (111)
%e A330469   ((1)(23))      ((1)(22))      ((1)(12))      ((1)(11))
%e A330469   ((2)(13))      ((2)(12))      ((2)(11))      ((1)(1)(1))
%e A330469   ((3)(12))      ((1)(2)(2))    ((1)(1)(2))    ((1)((1)(1)))
%e A330469   ((1)(2)(3))    ((1)((2)(2)))  ((1)((1)(2)))
%e A330469   ((1)((2)(3)))  ((2)((1)(2)))  ((2)((1)(1)))
%e A330469   ((2)((1)(3)))
%e A330469   ((3)((1)(2)))
%t A330469 allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
%t A330469 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A330469 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A330469 multing[t_,n_]:=Array[(t+#-1)/#&,n,1,Times];
%t A330469 amemo[m_]:=amemo[m]=1+Sum[Product[multing[amemo[s[[1]]],Length[s]],{s,Split[c]}],{c,Select[mps[m],Length[#]>1&]}];
%t A330469 Table[Sum[amemo[m],{m,allnorm[n]}],{n,0,5}]
%o A330469 (PARI)
%o A330469 EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
%o A330469 R(n, k)={my(v=[]); for(n=1, n, v=concat(v, EulerT(concat(v, [binomial(n+k-1, k-1)]))[n])); v}
%o A330469 seq(n)={concat([1], sum(k=1, n, R(n,k)*sum(r=k, n, binomial(r,k)*(-1)^(r-k))))} \\ _Andrew Howroyd_, Dec 29 2019
%Y A330469 The singleton-reduced version is A316651.
%Y A330469 The unlabeled version is A330465.
%Y A330469 The strongly normal case is A330467.
%Y A330469 The case when leaves are sets is A330764.
%Y A330469 Row sums of A330762.
%Y A330469 Cf. A000311, A000669, A004114, A005121, A005804, A141268, A292504, A292505, A316652, A318812, A318849, A319312, A330625.
%K A330469 nonn
%O A330469 0,3
%A A330469 _Gus Wiseman_, Dec 22 2019
%E A330469 Terms a(9) and beyond from _Andrew Howroyd_, Dec 29 2019