A327639 Number T(n,k) of proper k-times partitions of n; triangle T(n,k), n >= 0, 0 <= k <= max(0,n-1), read by rows.

1, 1, 1, 1, 1, 2, 1, 1, 4, 6, 3, 1, 6, 15, 16, 6, 1, 10, 45, 88, 76, 24, 1, 14, 93, 282, 420, 302, 84, 1, 21, 223, 1052, 2489, 3112, 1970, 498, 1, 29, 444, 2950, 9865, 18123, 18618, 10046, 2220, 1, 41, 944, 9030, 42787, 112669, 173338, 155160, 74938, 15108

Offset

0,6

Comments

In each step at least one part is replaced by the partition of itself into smaller parts. The parts are not resorted.

T(n,k) is defined for all n>=0 and k>=0. The triangle displays only positive terms. All other terms are zero.

Row n is the inverse binomial transform of the n-th row of array A323718.

Links

Alois P. Heinz, Rows n = 0..170, flattened

Wikipedia, Iverson bracket

Wikipedia, Partition (number theory)

Formula

T(n,k) = Sum_{i=0..k} (-1)^(k-i) * binomial(k,i) * A323718(n,i).

T(n,n-1) = A327631(n,n-1)/n = A327643(n) for n >= 1.

Sum_{k=1..n-1} k * T(n,k) = A327646(n).

Sum_{k=0..max(0,n-1)} (-1)^k * T(n,k) = [n<2], where [] is an Iverson bracket.

Example

T(4,0) = 1:  4
T(4,1) = 4:     T(4,2) = 6:          T(4,3) = 3:
  4-> 31          4-> 31 -> 211        4-> 31 -> 211 -> 1111
  4-> 22          4-> 31 -> 1111       4-> 22 -> 112 -> 1111
  4-> 211         4-> 22 -> 112        4-> 22 -> 211 -> 1111
  4-> 1111        4-> 22 -> 211
                  4-> 22 -> 1111
                  4-> 211-> 1111
Triangle T(n,k) begins:
  1;
  1;
  1,  1;
  1,  2,   1;
  1,  4,   6,    3;
  1,  6,  15,   16,     6;
  1, 10,  45,   88,    76,     24;
  1, 14,  93,  282,   420,    302,     84;
  1, 21, 223, 1052,  2489,   3112,   1970,    498;
  1, 29, 444, 2950,  9865,  18123,  18618,  10046,  2220;
  1, 41, 944, 9030, 42787, 112669, 173338, 155160, 74938, 15108;
  ...

Maple

b:= proc(n, i, k) option remember; `if`(n=0 or k=0, 1, `if`(i>1,
      b(n, i-1, k), 0) +b(i$2, k-1)*b(n-i, min(n-i, i), k))
    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..max(0, n-1)), n=0..12);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[n == 0 || k == 0, 1, If[i > 1, b[n, i - 1, k], 0] + b[i, i, k - 1] b[n - i, Min[n - i, i], k]];
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, Max[0, n - 1] }] // Flatten (* Jean-François Alcover, Dec 09 2020, after Alois P. Heinz *)

Crossrefs

Columns k=0-2 give A000012, A000065, A327769.

Row sums give A327644.

T(2n,n) gives A327645.

Cf. A323718, A327631, A327643, A327646.

Sequence in context: A351790 A322266 A190284 * A273891 A034870 A324224

Adjacent sequences: A327636 A327637 A327638 * A327640 A327641 A327642

Keyword

nonn,tabf

Author

Alois P. Heinz, Sep 20 2019

Status

approved