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%I #9 Feb 08 2020 20:41:42
%S 1,1,2,3,6,11,21,40,75,139,263,498,932,1761,3322,6244,11775,22204,
%T 41810,78795,148458,279690,527006,993033,1870881,3525109,6641904,
%U 12514243,23578708,44426222,83705148,157713617,297156310,559886943,1054911312,1987613556
%N Number of unlabeled rooted trees with n nodes where the multiplicities in the multiset of branches under any given node are distinct.
%H Andrew Howroyd, <a href="/A316796/b316796.txt">Table of n, a(n) for n = 1..100</a>
%e The a(6) = 11 trees:
%e (((((o)))))
%e ((((oo))))
%e (((ooo)))
%e (((o)(o)))
%e ((oo(o)))
%e ((oooo))
%e (oo((o)))
%e (oo(oo))
%e (o(o)(o))
%e (ooo(o))
%e (ooooo)
%t strut[n_]:=strut[n]=If[n===1,{{}},Select[Join@@Function[c,Union[Sort/@Tuples[strut/@c]]]/@IntegerPartitions[n-1],UnsameQ@@Length/@Split[#]&]];
%t Table[Length[strut[n]],{n,10}]
%o (PARI)
%o C(v,n)={my(recurse(r,b,p,k)=if(!r, 1, sum(m=1, r, if(!bittest(b,m), sum(i=1, min(r\m, p), my(f=if(i==p, k+1, 1)); if(v[i]>=f, (v[i]-f+1)*self()(r-m*i, bitor(b, 1<<m), i, f)/f)))))); recurse(n, 0, #v, 0)}
%o seq(n)={my(v=vector(n)); v[1]=1; for(n=2, #v, v[n]=C(v[1..n-1], n-1)); v} \\ _Andrew Howroyd_, Feb 08 2020
%Y Cf. A000081, A004111, A130091, A316793, A316794, A316795.
%K nonn
%O 1,3
%A _Gus Wiseman_, Jul 14 2018
%E Terms a(26) and beyond from _Andrew Howroyd_, Feb 08 2020
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