A319600 Number T(n,k) of plane partitions of n into parts of exactly k sorts; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 3, 4, 0, 6, 22, 18, 0, 13, 96, 198, 120, 0, 24, 330, 1272, 1800, 840, 0, 48, 1146, 7518, 19152, 20640, 7920, 0, 86, 3518, 36684, 148200, 274080, 234720, 75600, 0, 160, 10946, 177438, 1080960, 3083640, 4462560, 3180240, 887040, 0, 282, 32102, 788928, 6952440, 28621920, 62056080, 73175760, 44432640, 10886400

Offset: 0

Author

Alois P. Heinz, Sep 24 2018

Keywords

• nonn
• tabl

Examples

			Triangle T(n,k) begins:
  1;
  0,   1;
  0,   3,     4;
  0,   6,    22,     18;
  0,  13,    96,    198,     120;
  0,  24,   330,   1272,    1800,     840;
  0,  48,  1146,   7518,   19152,   20640,    7920;
  0,  86,  3518,  36684,  148200,  274080,  234720,   75600;
  0, 160, 10946, 177438, 1080960, 3083640, 4462560, 3180240, 887040;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..50, flattened
• Eric Weisstein's World of Mathematics, Plane partition
• Wikipedia, Plane partition

Crossrefs

Columns k=0-1 give: A000007, A000219 (for n>0).

Row sums give A319601.

Main diagonal gives A053529.

Cf. A255970, A319730.

Formula

T(n,k) = Sum_{i=0..k} (-1)^i * C(k,i) * A306100(n,k-i).

T(n,k) = k! * A319730(n,k).