OFFSET
0,3
COMMENTS
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.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..200
EXAMPLE
The a(3) = 24 trees:
(123) (122) (112) (111)
((1)(23)) ((1)(22)) ((1)(12)) ((1)(11))
((2)(13)) ((2)(12)) ((2)(11)) ((1)(1)(1))
((3)(12)) ((1)(2)(2)) ((1)(1)(2)) ((1)((1)(1)))
((1)(2)(3)) ((1)((2)(2))) ((1)((1)(2)))
((1)((2)(3))) ((2)((1)(2))) ((2)((1)(1)))
((2)((1)(3)))
((3)((1)(2)))
MATHEMATICA
allnorm[n_]:=If[n<=0, {{}}, Function[s, Array[Count[s, y_/; y<=#]+1&, n]]/@Subsets[Range[n-1]+1]];
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
multing[t_, n_]:=Array[(t+#-1)/#&, n, 1, Times];
amemo[m_]:=amemo[m]=1+Sum[Product[multing[amemo[s[[1]]], Length[s]], {s, Split[c]}], {c, Select[mps[m], Length[#]>1&]}];
Table[Sum[amemo[m], {m, allnorm[n]}], {n, 0, 5}]
PROG
(PARI)
EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
R(n, k)={my(v=[]); for(n=1, n, v=concat(v, EulerT(concat(v, [binomial(n+k-1, k-1)]))[n])); v}
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
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 22 2019
EXTENSIONS
Terms a(9) and beyond from Andrew Howroyd, Dec 29 2019
STATUS
approved