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%I #17 Sep 07 2019 06:03:34
%S 1,1,2,4,8,15,35,72,169,388,934,2234,5508,13557,33883,85017,215091,
%T 546496,1396524,3582383,9228470,23852918,61857180,160871716,419516462,
%U 1096671326,2873403980,7544428973,19847520789,52308750878,138095728065,365153263313,966978876376
%N Number of uniform rooted trees with n nodes.
%C An unlabeled rooted tree is uniform if the multiplicities of the branches directly under any given node are all equal.
%H Vaclav Kotesovec, <a href="/A317712/b317712.txt">Table of n, a(n) for n = 1..2250</a> (terms 1..200 from Andrew Howroyd)
%H Gus Wiseman, <a href="/A317712/a317712.png">All 72 uniform rooted trees with 8 nodes.</a>
%F a(n) ~ c * d^n / n^(3/2), where d = 2.774067238136373782458114960391469140405537808253... and c = 0.43338208953061974806801546569720246018271214... - _Vaclav Kotesovec_, Sep 07 2019
%e The a(5) = 8 uniform rooted trees:
%e ((((o))))
%e (((oo)))
%e ((o(o)))
%e ((ooo))
%e (o((o)))
%e (o(oo))
%e ((o)(o))
%e (oooo)
%t purt[n_]:=Join@@Table[Select[Union[Sort/@Tuples[purt/@ptn]],SameQ@@Length/@Split[#]&],{ptn,IntegerPartitions[n-1]}];
%t Table[Length[purt[n]],{n,10}]
%o (PARI) WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
%o seq(n)={my(v=[1]); for(n=2, n, my(t=WeighT(v)); v=concat(v, sumdiv(n-1, d, t[d]))); v} \\ _Andrew Howroyd_, Aug 28 2018
%Y Cf. A000081, A001190, A004111, A072774, A301700, A317588.
%Y Cf. A317705, A317707, A317708, A317709, A317710, A317711, A317717, A317718.
%K nonn
%O 1,3
%A _Gus Wiseman_, Aug 05 2018
%E Term a(21) and beyond from _Andrew Howroyd_, Aug 28 2018
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