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A321296 Number T(n,k) of colored set partitions of [n] where colors of the elements of subsets are in (weakly) increasing order and exactly k colors are used; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A321296 #29 Apr 30 2020 07:51:18
%S A321296 1,0,1,0,2,3,0,5,20,16,0,15,122,237,131,0,52,774,2751,3524,1496,0,203,
%T A321296 5247,30470,68000,65055,22482,0,877,38198,341244,1181900,1913465,
%U A321296 1462320,426833,0,4140,298139,3949806,19946654,48636035,61692855,39282229,9934563
%N A321296 Number T(n,k) of colored set partitions of [n] where colors of the elements of subsets are in (weakly) increasing order and exactly k colors are used; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H A321296 Alois P. Heinz, <a href="/A321296/b321296.txt">Rows n = 0..140, flattened</a>
%H A321296 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A321296 T(3,2) = 20: 1a2a3b, 1a2b3b, 1a|2a3b, 1a|2b3b, 1b|2a3a, 1b|2a3b, 1a3b|2a, 1b3b|2a, 1a3a|2b, 1a3b|2b, 1a2b|3a, 1b2b|3a, 1a2a|3b, 1a2b|3b, 1a|2a|3b, 1a|2b|3a, 1b|2a|3a, 1a|2b|3b, 1b|2a|3b, 1b|2b|3a.
%e A321296 Triangle T(n,k) begins:
%e A321296   1;
%e A321296   0,   1;
%e A321296   0,   2,     3;
%e A321296   0,   5,    20,     16;
%e A321296   0,  15,   122,    237,     131;
%e A321296   0,  52,   774,   2751,    3524,    1496;
%e A321296   0, 203,  5247,  30470,   68000,   65055,   22482;
%e A321296   0, 877, 38198, 341244, 1181900, 1913465, 1462320, 426833;
%e A321296   ...
%p A321296 A:= proc(n, k) option remember; `if`(n=0, 1, add(A(n-j, k)*
%p A321296       binomial(n-1, j-1)*binomial(k+j-1, j), j=1..n))
%p A321296     end:
%p A321296 T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):
%p A321296 seq(seq(T(n, k), k=0..n), n=0..10);
%t A321296 A[n_, k_] := A[n, k] = If[n == 0, 1, Sum[A[n-j, k] Binomial[n-1, j-1]* Binomial[k + j - 1, j], {j, n}]];
%t A321296 T[n_, k_] := Sum[A[n, k - i] (-1)^i Binomial[k, i], {i, 0, k}];
%t A321296 Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Apr 30 2020, after _Alois P. Heinz_ *)
%Y A321296 Columns k=0-2 give: A000007, A000110 (for n>0), A325890.
%Y A321296 Main diagonal gives A023998.
%Y A321296 Row sums give A325888.
%Y A321296 T(2n,n) gives A325889.
%Y A321296 Cf. A322670.
%K A321296 nonn,tabl
%O A321296 0,5
%A A321296 _Alois P. Heinz_, Aug 29 2019