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.

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

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