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A262078
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Number T(n,k) of partitions of an n-set with distinct block sizes and maximal block size equal to k; triangle T(n,k), k>=0, k<=n<=k*(k+1)/2, read by columns.
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4
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1, 1, 1, 3, 1, 4, 10, 60, 1, 5, 15, 140, 280, 1260, 12600, 1, 6, 21, 224, 630, 3780, 34650, 110880, 360360, 2522520, 37837800, 1, 7, 28, 336, 1050, 7392, 74844, 276276, 1513512, 9459450, 131171040, 428828400, 2058376320, 9777287520, 97772875200, 2053230379200
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OFFSET
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0,4
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LINKS
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EXAMPLE
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Triangle T(n,k) begins:
: 1;
: 1;
: 1;
: 3, 1;
: 4, 1;
: 10, 5, 1;
: 60, 15, 6, 1;
: 140, 21, 7, 1;
: 280, 224, 28, 8, 1;
: 1260, 630, 336, 36, 9, 1;
: 12600, 3780, 1050, 480, 45, 10, 1;
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MAPLE
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b:= proc(n, i) option remember; `if`(i*(i+1)/2<n, 0, `if`(n=0, 1,
b(n, i-1) +`if`(i>n, 0, binomial(n, i)*b(n-i, i-1))))
end:
T:= (n, k)-> b(n, k) -`if`(k=0, 0, b(n, k-1)):
seq(seq(T(n, k), n=k..k*(k+1)/2), k=0..7);
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MATHEMATICA
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b[n_, i_] := b[n, i] = If[i*(i+1)/2<n, 0, If[n==0, 1, b[n, i-1] + If[i>n, 0, Binomial[n, i]*b[n-i, i-1]]]]; T[n_, k_] := b[n, k] - If[k==0, 0, b[n, k-1]]; Table[T[n, k], {k, 0, 7}, {n, k, k*(k+1)/2}] // Flatten (* Jean-François Alcover, Dec 18 2016, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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