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

Author

Alois P. Heinz, Nov 04 2012

Keywords

• nonn
• tabl

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.

Examples

			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;
  ...
		

Links

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

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.

Programs

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 *)

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)).