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

0,5

Links

Alois P. Heinz, Rows n = 0..50, flattened

Eric Weisstein's World of Mathematics, Plane partition

Wikipedia, Plane partition

Formula

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

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

Example

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;

Crossrefs

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

Row sums give A319601.

Main diagonal gives A053529.

Cf. A255970, A319730.

Sequence in context: A105826 A110665 A063441 * A092894 A276563 A011338

Adjacent sequences: A319597 A319598 A319599 * A319601 A319602 A319603

Keyword

nonn,tabl

Author

Alois P. Heinz, Sep 24 2018

Status

approved