A119271 Triangle: number of exactly (m-1)-dimensional partitions of n, for n >= 1, m >= 0.

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 1, 5, 6, 1, 0, 1, 9, 18, 10, 1, 0, 1, 13, 44, 49, 15, 1, 0, 1, 20, 97, 172, 110, 21, 1, 0, 1, 28, 195, 512, 550, 216, 28, 1, 0, 1, 40, 377, 1370, 2195, 1486, 385, 36, 1, 0, 1, 54, 694, 3396, 7603, 7886, 3514, 638, 45, 1, 0, 1, 75, 1251, 7968

Offset: 1

Author

Franklin T. Adams-Watters, May 11 2006

Keywords

• nonn
• tabl
• hard

Comments

The partition of 1 is considered to be dimension -1 by convention.

Examples

			Table starts:
  1,
  0,1,
  0,1,1,
  0,1,3,1,
  0,1,5,6,1,
  ...
		

Links

• Suresh Govindarajan, Rows n = 1..26 of Triangle
• Suresh Govindarajan, Partitions Generator (gives partitions of integers <= 25 in any dimension using this triangle).
• Suresh Govindarajan, Refined counting of higher-dimensional partitions
• Suresh Govindarajan, Notes on higher-dimensional partitions, arXiv preprint arXiv:1203.4419 [math.CO], 2012.

Crossrefs

Cf. A119270, A096806. Column 1 is A007042.

Formula

a(n,m) = A096806(n,m-1)-a(n,m-1). Binomial transform of n-th row lists the (m-1) dimensional partitions of n.