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A318689 Number of powerful uniform rooted trees with n nodes.

%I A318689 #10 Dec 09 2020 15:39:30
%S A318689 1,1,2,3,5,6,11,12,19,23,35,36,63,64,98,112,173,174,291,292,473,509,
%T A318689 791,792,1345,1356,2158,2257,3634,3635,6053,6054,9807,10091,16173,
%U A318689 16216,26783,26784,43076,43880,70631,70632,114975,114976,184665,186996,298644,298645,481978,482011
%N A318689 Number of powerful uniform rooted trees with n nodes.
%C A318689 A powerful uniform rooted tree with n nodes is either a single powerful uniform branch with n-1 nodes, or a powerful uniform multiset (all multiplicities are equal to the same number > 1) of powerful uniform rooted trees with a total of n-1 nodes.
%H A318689 Andrew Howroyd, <a href="/A318689/b318689.txt">Table of n, a(n) for n = 1..1000</a>
%H A318689 Gus Wiseman, <a href="/A318689/a318689.png">All 35 powerful uniform rooted trees with 11 nodes.</a>
%e A318689 The a(8) = 12 powerful uniform rooted trees:
%e A318689   (((((((o)))))))
%e A318689   ((((((oo))))))
%e A318689   (((((o)(o)))))
%e A318689   ((((o))((o))))
%e A318689   (((((ooo)))))
%e A318689   (((o)(o)(o)))
%e A318689   ((((oooo))))
%e A318689   (((oo)(oo)))
%e A318689   ((oo(o)(o)))
%e A318689   (((ooooo)))
%e A318689   ((oooooo))
%e A318689   (ooooooo)
%t A318689 rurt[n_]:=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[rurt/@ptn]],Or[Length[#]==1,And[Min@@Length/@Split[#]>=2,SameQ@@Length/@Split[#]]]&],{ptn,IntegerPartitions[n-1]}]];
%t A318689 Table[Length[rurt[n]],{n,15}]
%o A318689 (PARI) WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
%o A318689 seq(n)={my(v=vector(n)); v[1]=1; for(n=1, n-1, my(u=WeighT(v[1..n])); v[n+1] = sumdiv(n,d,u[d]) - u[n] + v[n]); v} \\ _Andrew Howroyd_, Dec 09 2020
%Y A318689 Cf. A000081, A003238, A072774, A317705, A317707, A317710, A317717, A317718, A318611, A318612, A318690, A318691, A318692.
%K A318689 nonn
%O A318689 1,3
%A A318689 _Gus Wiseman_, Aug 31 2018
%E A318689 Terms a(21) and beyond from _Andrew Howroyd_, Dec 09 2020