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%I #78 Apr 30 2020 11:33:04
%S 1,0,1,0,1,4,0,1,18,30,0,1,74,360,360,0,1,310,3450,8880,6240,0,1,1382,
%T 31770,160080,271800,146160,0,1,6510,298662,2635920,8152200,10190880,
%U 4420080,0,1,32398,2918244,42687960,214527600,468669600,460474560,166924800
%N Number T(n,k) of colored set partitions of [n] where elements of subsets have distinct colors and exactly k colors are used; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H Alois P. Heinz, <a href="/A323128/b323128.txt">Rows n = 0..140, flattened</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e T(3,2) = 18: 1a|2a3b, 1a|2b3a, 1b|2a3b, 1b|2b3a, 1a3b|2a, 1b3a|2a, 1a3b|2b, 1b3a|2b, 1a2b|3a, 1b2a|3a, 1a2b|3b, 1b2a|3b, 1a|2a|3b, 1a|2b|3a, 1b|2a|3a, 1a|2b|3b, 1b|2a|3b, 1b|2b|3a.
%e Triangle T(n,k) begins:
%e 1;
%e 0, 1;
%e 0, 1, 4;
%e 0, 1, 18, 30;
%e 0, 1, 74, 360, 360;
%e 0, 1, 310, 3450, 8880, 6240;
%e 0, 1, 1382, 31770, 160080, 271800, 146160;
%e 0, 1, 6510, 298662, 2635920, 8152200, 10190880, 4420080;
%e ...
%p A:= proc(n, k) option remember; `if`(n=0, 1, add(k!/(k-j)!
%p *binomial(n-1, j-1)*A(n-j, k), j=1..min(k, n)))
%p end:
%p T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
%p seq(seq(T(n, k), k=0..n), n=0..10);
%t A[n_, k_] := A[n, k] = If[n==0, 1, Sum[k!/(k - j)! Binomial[n - 1, j - 1]* A[n - j, k], {j, Min[k, n]}]];
%t T[n_, k_] := Sum[A[n, k - i] (-1)^i Binomial[k, i], {i, 0, k}];
%t Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 30 2020, after _Alois P. Heinz_ *)
%Y Columns k=0-1 give: A000007, A057427.
%Y Row sums give A104600.
%Y Main diagonal gives A137341.
%Y T(2n,n) gives A324523.
%K nonn,tabl
%O 0,6
%A _Alois P. Heinz_, Aug 30 2019
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