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.
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, 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, 7, 12, 12, 8, 5, 3, 2, 1, 1, 1, 0, 1, 8, 17, 17, 13, 8, 5, 3, 2, 1, 1, 1
Offset: 1
Examples
Triangle begins:
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 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 7 12 12 8 5 3 2 1 1 1
0 1 8 17 17 13 8 5 3 2 1 1 1
0 1 10 25 26 20 14 8 5 3 2 1 1 1
0 1 12 34 39 31 21 14 8 5 3 2 1 1 1
The postpositive terms of row 9 {3, 4, 3, 2} count the following trees:
((ooooooo)) (((oooooo))) ((((ooooo)))) (((((oooo)))))
((o)(ooooo)) (((o)(oooo))) ((((o)(ooo)))) (((((o)(oo)))))
((oo)(oooo)) (((oo)(ooo))) ((((o))((oo))))
(((o))((ooo)))
Links
- Alois P. Heinz, Rows n = 1..200, flattened
- Gus Wiseman, Unlabeled balanced rooted semi-identity trees with 12 nodes, organized by depth.
Crossrefs
Programs
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Mathematica
ubk[n_,k_]:=Select[Join@@Table[Select[Union[Sort/@Tuples[ubk[#,k-1]&/@ptn]],UnsameQ@@DeleteCases[#,{}]&],{ptn,IntegerPartitions[n-1]}],SameQ[k,##]&@@Length/@Position[#,{}]&]; Table[Length[ubk[n,k]],{n,1,10},{k,0,n-1}]
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