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A318691 Number of series-reduced powerful uniform rooted trees with n nodes.

%I A318691 #9 Dec 09 2020 15:40:29
%S A318691 1,0,1,1,1,1,2,1,3,2,3,1,6,1,5,4,8,1,11,1,15,6,13,1,26,3,24,9,36,1,50,
%T A318691 1,58,14,67,7,107,1,105,25,160,1,213,1,245,45,291,1,443,5,492,68,644,
%U A318691 1,851,15,1019,106,1263,1,1785,1,1986,189,2592,26,3426,1,4071,292
%N A318691 Number of series-reduced powerful uniform rooted trees with n nodes.
%C A318691 A series-reduced powerful uniform rooted tree with n nodes is a powerful uniform multiset (all multiplicities are equal to the same number > 1) of series-reduced powerful uniform rooted trees with a total of n-1 nodes.
%H A318691 Andrew Howroyd, <a href="/A318691/b318691.txt">Table of n, a(n) for n = 1..1000</a>
%F A318691 a(p+1) = 1 for prime p. - _Andrew Howroyd_, Dec 09 2020
%e A318691 The a(19) = 11 series-reduced powerful uniform rooted trees with 19 nodes:
%e A318691   (((ooo)(ooo))((ooo)(ooo)))
%e A318691   ((oo(oo)(oo))(oo(oo)(oo)))
%e A318691   ((oo)(oo)(oo)(oo)(oo)(oo))
%e A318691   ((oo)(oo)(ooooo)(ooooo))
%e A318691   ((ooo)(ooo)(oooo)(oooo))
%e A318691   (oo(oo)(oo)(oooo)(oooo))
%e A318691   ((ooooo)(ooooo)(ooooo))
%e A318691   (ooo(oooo)(oooo)(oooo))
%e A318691   ((oooooooo)(oooooooo))
%e A318691   (oo(ooooooo)(ooooooo))
%e A318691   (oooooooooooooooooo)
%t A318691 rurt[n_]:=If[n==1,{{}},Join@@Table[Select[Union[Sort/@Tuples[rurt/@ptn]],And[Min@@Length/@Split[#]>=2,SameQ@@Length/@Split[#]]&],{ptn,IntegerPartitions[n-1]}]];
%t A318691 Table[Length[rurt[n]],{n,10}]
%o A318691 (PARI) WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v,n,(-1)^(n-1)/n))))-1,-#v)}
%o A318691 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} \\ _Andrew Howroyd_, Dec 09 2020
%Y A318691 Cf. A000081, A001190, A001678, A001694, A003238, A072774, A317705, A317707, A317710, A318611, A318612, A318689, A318692.
%K A318691 nonn
%O A318691 1,7
%A A318691 _Gus Wiseman_, Aug 31 2018
%E A318691 Terms a(51) and beyond from _Andrew Howroyd_, Dec 09 2020