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 A309973 Number T(n,k) of colored integer partitions of n using all colors of a k-set such that parts i have distinct color patterns in arbitrary order and each pattern for a part i has i colors in (weakly) increasing order; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 7
 1, 0, 1, 0, 1, 3, 0, 2, 6, 10, 0, 2, 21, 42, 47, 0, 3, 42, 177, 264, 246, 0, 4, 90, 619, 1746, 2095, 1602, 0, 5, 176, 1809, 7556, 16085, 16608, 11481, 0, 6, 348, 5211, 32621, 100030, 171480, 154385, 95503, 0, 8, 640, 13961, 120964, 522890, 1262832, 1842659, 1503232, 871030 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 LINKS Alois P. Heinz, Rows n = 0..140, flattened FORMULA Sum_{k=1..n} k * T(n,k) = A327680(n). EXAMPLE T(3,1) = 2: 3aaa, 2aa1a. T(3,2) = 6: 3aab, 3abb, 2aa1b, 2ab1a, 2ab1b, 2bb1a. T(3,3) = 10: 3abc, 2ab1c, 2ac1b, 2bc1a, 1a1b1c, 1a1c1b, 1b1a1c, 1b1c1a, 1c1a1b, 1c1b1a. Triangle T(n,k) begins:   1;   0, 1;   0, 1,   3;   0, 2,   6,   10;   0, 2,  21,   42,    47;   0, 3,  42,  177,   264,    246;   0, 4,  90,  619,  1746,   2095,   1602;   0, 5, 176, 1809,  7556,  16085,  16608,  11481;   0, 6, 348, 5211, 32621, 100030, 171480, 154385, 95503;   ... MAPLE b:= proc(n, i, k) option remember; `if`(n=0, 1,       `if`(i<1, 0, add(b(n-i*j, min(n-i*j, i-1), k)*        binomial(binomial(k+i-1, i), j)*j!, j=0..n/i)))     end: T:= (n, k)-> add(b(n\$2, i)*(-1)^(k-i)*binomial(k, i), i=0..k): seq(seq(T(n, k), k=0..n), n=0..12); MATHEMATICA b[n_, i_, k_] := b[n, i, k] = If[n==0, 1, If[i<1, 0, Sum[b[n - i j, Min[n - i j, i-1], k] Binomial[Binomial[k+i-1, i], j] j!, {j, 0, n/i}]]]; T[n_, k_] := Sum[b[n, n, i] (-1)^(k - i) Binomial[k, i], {i, 0, k}]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 29 2020, after Maple *) CROSSREFS Columns k=0-2 give: A000007, A000009 (for n>0), A327890. Main diagonal gives A005651. Row sums give A327679. T(2n,n) gives A327681. Cf. A327116, A327680. Sequence in context: A072328 A135040 A048733 * A298058 A298707 A274417 Adjacent sequences:  A309970 A309971 A309972 * A309974 A309975 A309976 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Sep 21 2019 STATUS approved

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Last modified September 30 18:12 EDT 2020. Contains 337440 sequences. (Running on oeis4.)