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

1,2

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.

Links

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

Formula

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

Example

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.

Prog

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

Crossrefs

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

See A306100 for a variant.

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

Sequence in context: A194232 A371567 A110042 * A364967 A337432 A123027

Adjacent sequences: A306098 A306099 A306100 * A306102 A306103 A306104

Keyword

nonn,tabl

Author

M. F. Hasler, Sep 22 2018

Status

approved