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%I #12 Sep 15 2018 02:05:22
%S 1,1,6,58,774,13171,272700,6655962,187172762,5959665653,211947272186,
%T 8327259067439,358211528524432,16744766791743136,845195057333580332,
%U 45814333121920927067,2654330505021077873594,163687811930206581162063,10705203621191765328300832
%N Number of series-reduced rooted identity trees with n leaves spanning an initial interval of positive integers.
%C 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.
%H Andrew Howroyd, <a href="/A316653/b316653.txt">Table of n, a(n) for n = 1..200</a>
%e The a(3) = 6 trees are (1(12)), (2(12)), (1(23)), (2(13)), (3(12)), (123).
%t sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t gro[m_]:=If[Length[m]==1,m,Select[Union[Sort/@Join@@(Tuples[gro/@#]&/@Select[mps[m],Length[#]>1&])],UnsameQ@@#&]];
%t allnorm[n_Integer]:=Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1];
%t Table[Sum[Length[gro[m]],{m,allnorm[n]}],{n,5}]
%o (PARI) \\ here R(n,2) is A031148.
%o WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
%o R(n,k)={my(v=[k]); for(n=2, n, v=concat(v, WeighT(concat(v,[0]))[n])); v}
%o seq(n)={sum(k=1, n, R(n,k)*sum(r=k, n, binomial(r,k)*(-1)^(r-k)) )} \\ _Andrew Howroyd_, Sep 14 2018
%Y Cf. A000081, A000311, A000669, A001678, A004111, A005804, A034691, A141268, A292504, A300660.
%Y Cf. A316651, A316652, A316654, A316655, A316656.
%K nonn
%O 1,3
%A _Gus Wiseman_, Jul 09 2018
%E Terms a(9) and beyond from _Andrew Howroyd_, Sep 14 2018
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