A244925 Number T(n,k) of n-node unlabeled rooted trees with every leaf at height k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.
1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 4, 3, 2, 1, 1, 0, 1, 4, 5, 3, 2, 1, 1, 0, 1, 7, 7, 6, 3, 2, 1, 1, 0, 1, 8, 12, 8, 6, 3, 2, 1, 1, 0, 1, 12, 18, 15, 9, 6, 3, 2, 1, 1, 0, 1, 14, 27, 23, 16, 9, 6, 3, 2, 1, 1, 0, 1, 21, 42, 39, 26, 17, 9, 6, 3, 2, 1, 1
Offset: 1
Examples
The A048816(5) = 5 rooted trees with 5 nodes with every leaf at the same height sorted by height are: : o : o o : o : o : : /( )\ : / \ | : | : | : : o o o o : o o o : o : o : : : | | /|\ : | : | : : : o o o o o : o : o : : : : / \ : | : : : : o o : o : : : : : | : : : : : o : : : : : : : ---1--- : -----2----- : --3-- : -4- : Thus row 5 = [0, 1, 2, 1, 1]. Triangle T(n,k) begins: 1; 0, 1; 0, 1, 1; 0, 1, 1, 1; 0, 1, 2, 1, 1; 0, 1, 2, 2, 1, 1; 0, 1, 4, 3, 2, 1, 1; 0, 1, 4, 5, 3, 2, 1, 1; 0, 1, 7, 7, 6, 3, 2, 1, 1; 0, 1, 8, 12, 8, 6, 3, 2, 1, 1; 0, 1, 12, 18, 15, 9, 6, 3, 2, 1, 1; 0, 1, 14, 27, 23, 16, 9, 6, 3, 2, 1, 1; ...
Links
- Alois P. Heinz, Rows n = 1..141, flattened
Crossrefs
Programs
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Maple
with(numtheory): T:= proc(n, k) option remember; `if`(n=1, 1, `if`(k=0, 0, add(add(`if`(d -
Mathematica
T[n_, k_] := T[n, k] = If[n == 1, 1, If[k == 0, 0, Sum[ Sum[ If[d