A306101 - OEIS

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

(list; table; graph; refs; listen; history; text; internal format)

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 ...]