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%I #16 Feb 02 2019 04:06:33
%S 1,0,1,0,1,1,0,1,1,1,0,1,1,1,1,0,1,2,1,1,1,0,1,2,2,1,1,1,0,1,3,3,2,1,
%T 1,1,0,1,3,4,3,2,1,1,1,0,1,5,6,5,3,2,1,1,1,0,1,5,9,7,5,3,2,1,1,1,0,1,
%U 7,12,12,8,5,3,2,1,1,1,0,1,8,17,17,13,8,5,3,2,1,1,1
%N Regular triangle read by rows where T(n,k) is the number of unlabeled balanced rooted semi-identity trees with n >= 1 nodes and depth 0 <= k < n.
%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. It is balanced if all leaves are the same distance from the root.
%H Alois P. Heinz, <a href="/A306269/b306269.txt">Rows n = 1..200, flattened</a>
%H Gus Wiseman, <a href="/A306269/a306269.png">Unlabeled balanced rooted semi-identity trees with 12 nodes, organized by depth.</a>
%e Triangle begins:
%e 1
%e 0 1
%e 0 1 1
%e 0 1 1 1
%e 0 1 1 1 1
%e 0 1 2 1 1 1
%e 0 1 2 2 1 1 1
%e 0 1 3 3 2 1 1 1
%e 0 1 3 4 3 2 1 1 1
%e 0 1 5 6 5 3 2 1 1 1
%e 0 1 5 9 7 5 3 2 1 1 1
%e 0 1 7 12 12 8 5 3 2 1 1 1
%e 0 1 8 17 17 13 8 5 3 2 1 1 1
%e 0 1 10 25 26 20 14 8 5 3 2 1 1 1
%e 0 1 12 34 39 31 21 14 8 5 3 2 1 1 1
%e The postpositive terms of row 9 {3, 4, 3, 2} count the following trees:
%e ((ooooooo)) (((oooooo))) ((((ooooo)))) (((((oooo)))))
%e ((o)(ooooo)) (((o)(oooo))) ((((o)(ooo)))) (((((o)(oo)))))
%e ((oo)(oooo)) (((oo)(ooo))) ((((o))((oo))))
%e (((o))((ooo)))
%t ubk[n_,k_]:=Select[Join@@Table[Select[Union[Sort/@Tuples[ubk[#,k-1]&/@ptn]],UnsameQ@@DeleteCases[#,{}]&],{ptn,IntegerPartitions[n-1]}],SameQ[k,##]&@@Length/@Position[#,{}]&];
%t Table[Length[ubk[n,k]],{n,1,10},{k,0,n-1}]
%Y Row sums give A306201.
%Y T(2n-1,n) gives A306274.
%Y Cf. A000081, A004111, A048816, A079500, A120803, A184155, A276625, A306200, A306202, A306203, A320222, A320270.
%K nonn,tabl
%O 1,18
%A _Gus Wiseman_, Feb 01 2019
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