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A276922
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Number T(n,k) of ordered set partitions of [n] where the maximal block size equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
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14
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1, 0, 1, 0, 2, 1, 0, 6, 6, 1, 0, 24, 42, 8, 1, 0, 120, 330, 80, 10, 1, 0, 720, 2970, 860, 120, 12, 1, 0, 5040, 30240, 10290, 1540, 168, 14, 1, 0, 40320, 345240, 136080, 21490, 2464, 224, 16, 1, 0, 362880, 4377240, 1977360, 326970, 38808, 3696, 288, 18, 1
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OFFSET
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0,5
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LINKS
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FORMULA
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E.g.f. for column k>0: 1/(1-Sum_{i=1..k} x^i/i!) - 1/(1-Sum_{i=1..k-1} x^i/i!).
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EXAMPLE
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Triangle T(n,k) begins:
1;
0, 1;
0, 2, 1;
0, 6, 6, 1;
0, 24, 42, 8, 1;
0, 120, 330, 80, 10, 1;
0, 720, 2970, 860, 120, 12, 1;
0, 5040, 30240, 10290, 1540, 168, 14, 1;
0, 40320, 345240, 136080, 21490, 2464, 224, 16, 1;
...
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MAPLE
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A:= proc(n, k) option remember; `if`(n=0, 1, add(
A(n-i, k)*binomial(n, i), i=1..min(n, k)))
end:
T:= (n, k)-> A(n, k) -`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=0..n), n=0..10);
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MATHEMATICA
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A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[A[n - i, k]*Binomial[n, i], {i, 1, Min[n, k]}]]; T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]]; Table[T[n, k], {n, 0, 10}, { k, 0, n}] // Flatten (* Jean-François Alcover, Feb 11 2017, translated from Maple *)
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CROSSREFS
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Columns k=0-10 give: A000007, A000142 (for n>0), A320758, A320759, A320760, A320761, A320762, A320763, A320764, A320765, A320766.
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KEYWORD
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AUTHOR
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STATUS
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approved
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