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A306200 Number of unlabeled rooted semi-identity trees with n nodes.

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