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A306101
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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.
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7
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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Alois P. Heinz, Antidiagonals n = 1..50, flattened
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FORMULA
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T(n,k) = Sum_{j=1..n} A091298(n,j)*k^j.
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EXAMPLE
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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.
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PROG
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(PARI) A306101(n, k)=sum(j=1, n, A091298(n, j)*k^j)
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CROSSREFS
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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: A340914 A194232 A110042 * A337432 A123027 A100652
Adjacent sequences: A306098 A306099 A306100 * A306102 A306103 A306104
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KEYWORD
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nonn,tabl
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AUTHOR
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M. F. Hasler, Sep 22 2018
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STATUS
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approved
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