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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. 3
 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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,18 COMMENTS 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. LINKS Alois P. Heinz, Rows n = 1..200, flattened Gus Wiseman, Unlabeled balanced rooted semi-identity trees with 12 nodes, organized by depth. EXAMPLE 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))) 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}] CROSSREFS Row sums give A306201. T(2n-1,n) gives A306274. Cf. A000081, A004111, A048816, A079500, A120803, A184155, A276625, A306200, A306202, A306203, A320222, A320270. Sequence in context: A278287 A227752 A316771 * A035212 A318133 A068029 Adjacent sequences: A306266 A306267 A306268 * A306270 A306271 A306272 KEYWORD nonn,tabl AUTHOR Gus Wiseman, Feb 01 2019 STATUS approved

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Last modified October 2 23:30 EDT 2023. Contains 365841 sequences. (Running on oeis4.)