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A306201 Number of unlabeled balanced rooted semi-identity trees with n nodes.

%I A306201 #13 Feb 02 2019 03:46:00
%S A306201 0,1,1,2,3,4,6,8,12,16,25,35,53,77,117,173,265,396,605,919,1408,2147,
%T A306201 3305,5070,7819,12049,18635,28811,44672,69264,107618,167292,260446,
%U A306201 405686,632743,987441,1542555,2411208,3772247,5905002,9250436,14499234,22740910,35686092
%N A306201 Number of unlabeled balanced rooted semi-identity trees with n nodes.
%C A306201 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. It is balanced if all leaves are the same distance from the root. The only balanced identity trees are rooted paths.
%H A306201 Alois P. Heinz, <a href="/A306201/b306201.txt">Table of n, a(n) for n = 0..500</a>
%e A306201 The a(1) = 1 through a(7) = 8 balanced rooted semi-identity trees:
%e A306201   o  (o)  (oo)   (ooo)    (oooo)     (ooooo)      (oooooo)
%e A306201           ((o))  ((oo))   ((ooo))    ((oooo))     ((ooooo))
%e A306201                  (((o)))  (((oo)))   (((ooo)))    (((oooo)))
%e A306201                           ((((o))))  ((o)(oo))    ((o)(ooo))
%e A306201                                      ((((oo))))   ((((ooo))))
%e A306201                                      (((((o)))))  (((o)(oo)))
%e A306201                                                   (((((oo)))))
%e A306201                                                   ((((((o))))))
%t A306201 ursit[n_]:=Join@@Table[Select[Union[Sort/@Tuples[ursit/@ptn]],UnsameQ@@DeleteCases[#,{}]&],{ptn,IntegerPartitions[n-1]}];
%t A306201 Table[Length[Select[ursit[n],SameQ@@Length/@Position[#,{}]&]],{n,10}]
%Y A306201 Cf. A000081, A004111, A048816, A079500, A120803, A306200, A316624, A317712, A320169, A320270.
%Y A306201 Row sums of A306269.
%K A306201 nonn
%O A306201 0,4
%A A306201 _Gus Wiseman_, Jan 29 2019
%E A306201 More terms from _Alois P. Heinz_, Jan 29 2019