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A244530
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Number T(n,k) of ordered unlabeled rooted trees with n nodes such that the minimal outdegree of inner nodes equals k; triangle T(n,k), n>=1, 0<=k<=n-1, read by rows.
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11
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1, 0, 1, 0, 1, 1, 0, 4, 0, 1, 0, 11, 2, 0, 1, 0, 36, 5, 0, 0, 1, 0, 117, 11, 3, 0, 0, 1, 0, 393, 28, 7, 0, 0, 0, 1, 0, 1339, 78, 8, 4, 0, 0, 0, 1, 0, 4630, 201, 21, 9, 0, 0, 0, 0, 1, 0, 16193, 532, 55, 10, 5, 0, 0, 0, 0, 1, 0, 57201, 1441, 121, 11, 11, 0, 0, 0, 0, 0, 1
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OFFSET
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1,8
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COMMENTS
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T(1,0) = 1 by convention.
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LINKS
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EXAMPLE
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T(5,1) = 11:
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o o o o o o o
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o
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 4, 0, 1;
0, 11, 2, 0, 1;
0, 36, 5, 0, 0, 1;
0, 117, 11, 3, 0, 0, 1;
0, 393, 28, 7, 0, 0, 0, 1;
0, 1339, 78, 8, 4, 0, 0, 0, 1;
0, 4630, 201, 21, 9, 0, 0, 0, 0, 1;
0, 16193, 532, 55, 10, 5, 0, 0, 0, 0, 1;
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MAPLE
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b:= proc(n, t, k) option remember; `if`(n=0,
`if`(t in [0, k], 1, 0), `if`(t>n, 0, add(b(j-1, k$2)*
b(n-j, max(0, t-1), k), j=1..n)))
end:
T:= (n, k)-> b(n-1, k$2) -`if`(n=1 and k=0, 0, b(n-1, k+1$2)):
seq(seq(T(n, k), k=0..n-1), n=1..14);
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MATHEMATICA
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b[n_, t_, k_] := b[n, t, k] = If[n == 0, If[t == 0 || t == k, 1, 0], If[t>n, 0, Sum[b[j-1, k, k]*b[n-j, Max[0, t-1], k], {j, 1, n}]]]; T[n_, k_] := b[n-1, k, k] - If[n == 1 && k == 0, 0, b[n-1, k+1, k+1]]; Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 14}] // Flatten (* Jean-François Alcover, Jan 13 2015, translated from Maple *)
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CROSSREFS
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Columns k=0-10 give: A063524, A106640(n-2), A244531, A244532, A244533, A244534, A244535, A244536, A244537, A244538, A244539.
Cf. A244454 (unordered unlabeled rooted trees).
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KEYWORD
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AUTHOR
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STATUS
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approved
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