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A306269 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.

%I A306269 #16 Feb 02 2019 04:06:33
%S A306269 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 A306269 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 A306269 7,12,12,8,5,3,2,1,1,1,0,1,8,17,17,13,8,5,3,2,1,1,1
%N A306269 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 A306269 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 A306269 Alois P. Heinz, <a href="/A306269/b306269.txt">Rows n = 1..200, flattened</a>
%H A306269 Gus Wiseman, <a href="/A306269/a306269.png">Unlabeled balanced rooted semi-identity trees with 12 nodes, organized by depth.</a>
%e A306269 Triangle begins:
%e A306269   1
%e A306269   0  1
%e A306269   0  1  1
%e A306269   0  1  1  1
%e A306269   0  1  1  1  1
%e A306269   0  1  2  1  1  1
%e A306269   0  1  2  2  1  1  1
%e A306269   0  1  3  3  2  1  1  1
%e A306269   0  1  3  4  3  2  1  1  1
%e A306269   0  1  5  6  5  3  2  1  1  1
%e A306269   0  1  5  9  7  5  3  2  1  1  1
%e A306269   0  1  7 12 12  8  5  3  2  1  1  1
%e A306269   0  1  8 17 17 13  8  5  3  2  1  1  1
%e A306269   0  1 10 25 26 20 14  8  5  3  2  1  1  1
%e A306269   0  1 12 34 39 31 21 14  8  5  3  2  1  1  1
%e A306269 The postpositive terms of row 9 {3, 4, 3, 2} count the following trees:
%e A306269   ((ooooooo))   (((oooooo)))    ((((ooooo))))    (((((oooo)))))
%e A306269   ((o)(ooooo))  (((o)(oooo)))   ((((o)(ooo))))   (((((o)(oo)))))
%e A306269   ((oo)(oooo))  (((oo)(ooo)))   ((((o))((oo))))
%e A306269                 (((o))((ooo)))
%t A306269 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 A306269 Table[Length[ubk[n,k]],{n,1,10},{k,0,n-1}]
%Y A306269 Row sums give A306201.
%Y A306269 T(2n-1,n) gives A306274.
%Y A306269 Cf. A000081, A004111, A048816, A079500, A120803, A184155, A276625, A306200, A306202, A306203, A320222, A320270.
%K A306269 nonn,tabl
%O A306269 1,18
%A A306269 _Gus Wiseman_, Feb 01 2019