A218698 Number T(n,k) of ways to divide the partitions of n into nonempty consecutive subsequences each of which contains only equal parts and parts from distinct subsequences differ by at least k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 1, 1, 3, 2, 2, 6, 3, 2, 2, 14, 5, 4, 3, 3, 27, 7, 4, 3, 2, 2, 60, 11, 8, 6, 5, 4, 4, 117, 15, 8, 6, 4, 3, 2, 2, 246, 22, 13, 9, 8, 6, 5, 4, 4, 490, 30, 15, 12, 8, 7, 5, 4, 3, 3, 1002, 42, 22, 14, 12, 9, 8, 6, 5, 4, 4, 1998, 56, 24, 16, 12, 10, 7, 6, 4, 3, 2, 2

Offset

0,4

Comments

T(n,k) is defined for n,k >= 0. The triangle contains terms with k <= n. T(n,k) = T(n,n) = A000005(n) for k >= n. For k>0: T(n,k) = number of partitions of n in which any two distinct parts differ by at least k, or, equivalently, T(n,k) = number of partitions of n in which each part, with the possible exception of the largest, occurs at least k times.

Links

Alois P. Heinz, Rows n = 0..140, flattened

Formula

G.f. of column k: 1 + Sum_{j>=1} x^j/(1-x^j) * Product_{i=1..j-1} (1+x^(k*i)/(1-x^i)).

Example

T(4,0) = 14: [[1],[1],[1],[1]], [[1,1],[1],[1]], [[1],[1,1],[1]], [[1,1,1],[1]], [[1],[1],[1,1]], [[1,1],[1,1]], [[1],[1,1,1]], [[1,1,1,1]], [[1],[1],[2]], [[1,1],[2]], [[2],[2]], [[2,2]], [[1],[3]], [[4]].
T(4,1) = 5: [[1,1,1,1]], [[1,1],[2]], [[2,2]], [[1],[3]], [[4]].
T(4,2) = 4: [[1,1,1,1]], [[2,2]], [[1],[3]], [[4]].
T(4,3) = T(4,4) = A000005(4) = 3: [[1,1,1,1]], [[2,2]], [[4]].
Triangle T(n,k) begins:
    1;
    1,  1;
    3,  2,  2;
    6,  3,  2,  2;
   14,  5,  4,  3,  3;
   27,  7,  4,  3,  2,  2;
   60, 11,  8,  6,  5,  4,  4;
  117, 15,  8,  6,  4,  3,  2,  2;
  ...

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0,
       b(n, i-1, k) +add(b(n-i*j, i-k, k), j=1..n/i)))
    end:
T:= (n, k)-> b(n, n, 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,  b[n, i - 1, k] + Sum[b[n - i*j, i - k, k], {j, 1, n/i}]]]; T[n_, k_] := b[n, n, k]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 27 2013, translated from Maple *)

Crossrefs

Columns k=0-10 give: A006951, A000041, A116931, A116932, A218699, A218700, A218701, A218702, A218703, A218704, A218705.

Main diagonal gives: A000005.

T(2n,n) gives A319776.

Sequence in context: A093055 A285733 A106335 * A352836 A065474 A272332

Adjacent sequences: A218695 A218696 A218697 * A218699 A218700 A218701

Keyword

nonn,tabl

Author

Alois P. Heinz, Nov 04 2012

Status

approved