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

0,33

Links

Table of n, a(n) for n=0..90.

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

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.

Example

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

Mathematica

T[n_, m_] := T[n, m] = Which[n==m, 1, m<n/2 || m>n, 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 *)

Crossrefs

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

Sequence in context: A086009 A086010 A321929 * A059607 A176724 A015318

Adjacent sequences: A089195 A089196 A089197 * A089199 A089200 A089201

Keyword

nonn,tabl

Author

N. J. A. Sloane, Dec 10 2003

Status

approved