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%I #17 May 10 2021 07:40:41
%S 0,1,1,2,4,8,18,41,98,237,591,1488,3805,9820,25593,67184,177604,
%T 472177,1261998,3388434,9136019,24724904,67141940,182892368,499608724,
%U 1368340326,3756651116,10336434585,28499309291,78727891420,217870037932,603934911859,1676720329410
%N Number of unlabeled rooted semi-identity trees with n nodes.
%C A rooted tree is a semi-identity tree if the non-leaf branches of the root are all distinct and are themselves semi-identity trees.
%H Alois P. Heinz, <a href="/A306200/b306200.txt">Table of n, a(n) for n = 0..2166</a>
%e The a(1) = 1 through a(7) = 8 trees:
%e o (o) (oo) (ooo) (oooo) (ooooo)
%e ((o)) ((oo)) ((ooo)) ((oooo))
%e (o(o)) (o(oo)) (o(ooo))
%e (((o))) (oo(o)) (oo(oo))
%e (((oo))) (ooo(o))
%e ((o(o))) (((ooo)))
%e (o((o))) ((o)(oo))
%e ((((o)))) ((o(oo)))
%e ((oo(o)))
%e (o((oo)))
%e (o(o(o)))
%e (oo((o)))
%e ((((oo))))
%e (((o(o))))
%e ((o)((o)))
%e ((o((o))))
%e (o(((o))))
%e (((((o)))))
%p b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
%p add(b(n-i*j, i-1)*binomial(a(i), j), j=0..n/i))
%p end:
%p a:= n-> `if`(n=0, 0, b(n-1$2)):
%p seq(a(n), n=0..35); # _Alois P. Heinz_, Jan 29 2019
%t ursit[n_]:=Join@@Table[Select[Union[Sort/@Tuples[ursit/@ptn]],UnsameQ@@DeleteCases[#,{}]&],{ptn,IntegerPartitions[n-1]}];
%t Table[Length[ursit[n]],{n,10}]
%t (* Second program: *)
%t b[n_, i_] := b[n, i] = If[n == 0 || i == 1, 1,
%t Sum[b[n - i*j, i - 1]*Binomial[a[i], j], {j, 0, n/i}]];
%t a[n_] := If[n == 0, 0, b[n - 1, n - 1]];
%t a /@ Range[0, 35] (* _Jean-François Alcover_, May 10 2021, after _Alois P. Heinz_ *)
%Y Cf. A000081, A004111, A276625, A301700, A306201, A316471, A316474, A317708, A317712, A317718.
%K nonn
%O 0,4
%A _Gus Wiseman_, Jan 29 2019
%E More terms from _Alois P. Heinz_, Jan 29 2019
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