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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. 6
 1, 0, 1, 0, 2, 3, 0, 5, 20, 16, 0, 15, 122, 237, 131, 0, 52, 774, 2751, 3524, 1496, 0, 203, 5247, 30470, 68000, 65055, 22482, 0, 877, 38198, 341244, 1181900, 1913465, 1462320, 426833, 0, 4140, 298139, 3949806, 19946654, 48636035, 61692855, 39282229, 9934563 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 LINKS Alois P. Heinz, Rows n = 0..140, flattened Wikipedia, Partition of a set EXAMPLE 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. Triangle T(n,k) begins:   1;   0,   1;   0,   2,     3;   0,   5,    20,     16;   0,  15,   122,    237,     131;   0,  52,   774,   2751,    3524,    1496;   0, 203,  5247,  30470,   68000,   65055,   22482;   0, 877, 38198, 341244, 1181900, 1913465, 1462320, 426833;   ... MAPLE A:= proc(n, k) option remember; `if`(n=0, 1, add(A(n-j, k)*       binomial(n-1, j-1)*binomial(k+j-1, j), j=1..n))     end: T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k): seq(seq(T(n, k), k=0..n), n=0..10); MATHEMATICA 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[n_, k_] := Sum[A[n, k - i] (-1)^i Binomial[k, i], {i, 0, k}]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 30 2020, after Alois P. Heinz *) CROSSREFS Columns k=0-2 give: A000007, A000110 (for n>0), A325890. Main diagonal gives A023998. Row sums give A325888. T(2n,n) gives A325889. Cf. A322670. Sequence in context: A004179 A324640 A122830 * A190902 A115562 A127468 Adjacent sequences:  A321293 A321294 A321295 * A321297 A321298 A321299 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Aug 29 2019 STATUS approved

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Last modified May 17 15:31 EDT 2021. Contains 343980 sequences. (Running on oeis4.)