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%I #25 Dec 08 2020 17:20:42
%S 1,0,1,0,2,4,0,5,30,30,0,15,210,540,360,0,52,1560,7800,12480,6240,0,
%T 203,12586,109620,316680,365400,146160,0,877,110502,1583862,7366800,
%U 14733600,13260240,4420080,0,4140,1051560,23995440,169011360,521640000,792892800,584236800,166924800
%N Number T(n,k) of colored set partitions of [n] where exactly k colors are used for the elements; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H Alois P. Heinz, <a href="/A323099/b323099.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>
%F T(n,k) = Bell(n) * Sum_{i=0..k} (k-i)^n * (-1)^i * C(k,i).
%F T(n,k) = Bell(n) * A131689(n,k).
%F T(n,k) = Bell(n) * Stirling2(n,k) * k!.
%e Triangle T(n,k) begins:
%e 1;
%e 0, 1;
%e 0, 2, 4;
%e 0, 5, 30, 30;
%e 0, 15, 210, 540, 360;
%e 0, 52, 1560, 7800, 12480, 6240;
%e 0, 203, 12586, 109620, 316680, 365400, 146160;
%e 0, 877, 110502, 1583862, 7366800, 14733600, 13260240, 4420080;
%e ...
%p A:= proc(n, k) option remember; `if`(n=0, 1, add(
%p A(n-j, k)*binomial(n-1, j-1)*k^j, j=1..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);
%p # second Maple program:
%p T:= (n, k)-> combinat[bell](n)*Stirling2(n,k)*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[A[n-j, k] Binomial[n-1, j-1] k^j, {j, 1, n}]];
%t T[n_, k_] := Sum[A[n, k - i] (-1)^i Binomial[k, i], {i, 0, k}];
%t Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten
%t (* second program: *)
%t T[n_, k_] := BellB[n] StirlingS2[n, k] k!;
%t Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Dec 08 2020, after _Alois P. Heinz_ *)
%Y Columns k=0-1 give: A000007, A000110 (for n>0).
%Y Row sums give A121017.
%Y Main diagonal gives A137341.
%Y Cf. A000142, A008277, A048993, A019538, A131689.
%K nonn,tabl
%O 0,5
%A _Alois P. Heinz_, Aug 30 2019
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