A316653 Number of series-reduced rooted identity trees with n leaves spanning an initial interval of positive integers.

1, 1, 6, 58, 774, 13171, 272700, 6655962, 187172762, 5959665653, 211947272186, 8327259067439, 358211528524432, 16744766791743136, 845195057333580332, 45814333121920927067, 2654330505021077873594, 163687811930206581162063, 10705203621191765328300832

Offset: 1

Author

Gus Wiseman, Jul 09 2018

Keywords

• nonn

Comments

A rooted tree is series-reduced if every non-leaf node has at least two branches. It is an identity tree if no branch appears multiple times under the same root.

Examples

			The a(3) = 6 trees are (1(12)), (2(12)), (1(23)), (2(13)), (3(12)), (123).
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..200

Crossrefs

Cf. A000081, A000311, A000669, A001678, A004111, A005804, A034691, A141268, A292504, A300660.

Cf. A316651, A316652, A316654, A316655, A316656.

Programs

Mathematica

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]]]];
gro[m_]:=If[Length[m]==1,m,Select[Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])],UnsameQ@@#&]];
allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
Table[Sum[Length[gro[m]],{m,allnorm[n]}],{n,5}]

PARI

\\ here R(n,2) is A031148.
WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
R(n,k)={my(v=[k]); for(n=2, n, v=concat(v, WeighT(concat(v,[0]))[n])); v}
seq(n)={sum(k=1, n, R(n,k)*sum(r=k, n, binomial(r,k)*(-1)^(r-k)) )} \\ Andrew Howroyd, Sep 14 2018

Extensions

Terms a(9) and beyond from Andrew Howroyd, Sep 14 2018