A306101 Square array T(n,k) = number of plane partitions of n with parts colored in (at most) k colors; n, k >= 1; read by antidiagonals.

1, 2, 3, 3, 10, 6, 4, 21, 34, 13, 5, 36, 102, 122, 24, 6, 55, 228, 525, 378, 48, 7, 78, 430, 1540, 2334, 1242, 86, 8, 105, 726, 3605, 8964, 11100, 3690, 160, 9, 136, 1134, 7278, 25980, 56292, 47496, 11266, 282, 10, 171, 1672, 13237, 62574, 203280, 316388, 210756, 32666, 500, 11, 210, 2358, 22280, 132258, 586878, 1417530

Offset: 1

Author

M. F. Hasler, Sep 22 2018

Keywords

• nonn
• tabl

Comments

One could have included a row 0 with all 1's, since there is exactly one partition of n = 0, the empty sum, for which all terms (since there are none) are colored in one among k colors.

Examples

			The array starts:
  [      1       2       3       4       5 ...] = A000027
  [      3      10      21      36      55 ...] = A014105
  [      6      34     102     228     430 ...] = A067389
  [     13     122     525    1540    3605 ...]
  [     24     378    2334    8964   25980 ...]
  [     48    1242   11100   56292  203280 ...]
   A000219 A306099 A306093 A306094 A306094
For concrete examples, see A306099 and A306093.
		

Links

• Alois P. Heinz, Antidiagonals n = 1..50, flattened

Crossrefs

Cf. A091298, A208447, A000219, A000027, A014105, A067389.

See A306100 for a variant.

Cf. A000219, A306099, A306093, A306094, A306095 for columns 1..5.

Programs

PARI

A306101(n,k)=sum(j=1,n,A091298(n,j)*k^j)

Formula

T(n,k) = Sum_{j=1..n} A091298(n,j)*k^j.