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A261780
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Number A(n,k) of compositions of n where each part i is marked with a word of length i over a k-ary alphabet whose letters appear in alphabetical order; square array A(n,k), n>=0, k>=0, read by antidiagonals.
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14
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1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 7, 4, 0, 1, 4, 15, 24, 8, 0, 1, 5, 26, 73, 82, 16, 0, 1, 6, 40, 164, 354, 280, 32, 0, 1, 7, 57, 310, 1031, 1716, 956, 64, 0, 1, 8, 77, 524, 2395, 6480, 8318, 3264, 128, 0, 1, 9, 100, 819, 4803, 18501, 40728, 40320, 11144, 256, 0
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OFFSET
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0,8
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COMMENTS
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Also the number of k-compositions of n: matrices with k rows of nonnegative integers with positive column sums and total element sum n.
A(2,2) = 7: (matrices and corresponding marked compositions are given)
[1 1] [0 0] [1 0] [0 1] [1] [2] [0]
[0 0] [1 1] [0 1] [1 0] [1] [0] [2]
1a1a, 1b1b, 1a1b, 1b1a, 2ab, 2aa, 2bb.
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LINKS
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E. Munarini, M. Poneti, S. Rinaldi, Matrix compositions, Journal of Integer Sequences, Vol. 12 (2009), Article 09.4.8
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FORMULA
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G.f. of column k: (1-x)^k/(2*(1-x)^k-1).
A(n,k) = Sum_{i=0..k} C(k,i) * A261781(n,k-i).
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EXAMPLE
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A(3,2) = 24: 3aaa, 3aab, 3abb, 3bbb, 2aa1a, 2aa1b, 2ab1a, 2ab1b, 2bb1a, 2bb1b, 1a2aa, 1a2ab, 1a2bb, 1b2aa, 1b2ab, 1b2bb, 1a1a1a, 1a1a1b, 1a1b1a, 1a1b1b, 1b1a1a, 1b1a1b, 1b1b1a, 1b1b1b.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, ...
0, 2, 7, 15, 26, 40, 57, ...
0, 4, 24, 73, 164, 310, 524, ...
0, 8, 82, 354, 1031, 2395, 4803, ...
0, 16, 280, 1716, 6480, 18501, 44022, ...
0, 32, 956, 8318, 40728, 142920, 403495, ...
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MAPLE
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A:= proc(n, k) option remember; `if`(n=0, 1,
add(A(n-j, k)*binomial(j+k-1, k-1), j=1..n))
end:
seq(seq(A(n, d-n), n=0..d), d=0..12);
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MATHEMATICA
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a[n_, k_] := SeriesCoefficient[(1-x)^k/(2*(1-x)^k-1), {x, 0, n}]; Table[ a[n-k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Feb 07 2017 *)
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CROSSREFS
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Columns k=0-10 give: A000007, A011782, A003480, A145839, A145840, A145841, A161434, A261799, A261800, A261801, A261802.
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KEYWORD
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AUTHOR
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STATUS
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approved
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