A089198 Triangle read by rows: T(n,k) (n>=0, 0<=k<=n) = number of non-squashing partitions of n into distinct parts of which the greatest is k.

1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 2, 1, 1, 1, 0, 0, 0, 0, 1, 2, 1, 1, 1, 0, 0, 0, 0, 0, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 2, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 3, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 3, 3, 2, 2, 1, 1, 1

Offset: 0

Author

N. J. A. Sloane, Dec 10 2003

Keywords

• nonn
• tabl

Examples

			Triangle begins:
1
0 1
0 0 1
0 0 1 1
0 0 0 1 1
0 0 0 1 1 1
0 0 0 1 1 1 1
0 0 0 0 2 1 1 1
0 0 0 0 1 2 1 1 1
		

Links

• N. J. A. Sloane and J. A. Sellers, On non-squashing partitions, Discrete Math., 294 (2005), 259-274.

Crossrefs

Row sums = A088567. Rows read from right to left also give (essentially) A088567.

Programs

Mathematica

T[n_, m_] := T[n, m] = Which[n==m, 1, mn, 0, True, Sum[T[n-m, i], {i, 0, m-1}]];
Table[T[n, m], {n, 0, 12}, {m, 0, n}] // Flatten (* Jean-François Alcover, Feb 13 2019 *)

Formula

The nonzero values of T(n, m) lie within a certain cone: T(n, m) = 0 if m < n/2 or if m > n. For m <= n <= 2m, T(n, m) = sum_{i=0}^{m-1} T(n-m, i).

For m <= n <= 2m, T(n, m) = b(n-m) if n < 2m, = b(n-m) - 1 if n = 2m, where b = A088567.