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A306713
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Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of g.f. 1/(1-x^k-x^(k+1)).
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5
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1, 1, 1, 1, 0, 2, 1, 0, 1, 3, 1, 0, 0, 1, 5, 1, 0, 0, 1, 1, 8, 1, 0, 0, 0, 1, 2, 13, 1, 0, 0, 0, 1, 0, 2, 21, 1, 0, 0, 0, 0, 1, 1, 3, 34, 1, 0, 0, 0, 0, 1, 0, 2, 4, 55, 1, 0, 0, 0, 0, 0, 1, 0, 1, 5, 89, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 7, 144, 1, 0, 0, 0, 0, 0, 0, 1, 0, 2, 3, 9, 233
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OFFSET
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0,6
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COMMENTS
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A(n,k) is the number of compositions of n into parts k and k+1.
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LINKS
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FORMULA
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A(n,k) = Sum_{j=0..floor(n/k)} binomial(j,n-k*j).
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 0, 0, 0, 0, 0, 0, 0, 0, ...
2, 1, 0, 0, 0, 0, 0, 0, 0, ...
3, 1, 1, 0, 0, 0, 0, 0, 0, ...
5, 1, 1, 1, 0, 0, 0, 0, 0, ...
8, 2, 0, 1, 1, 0, 0, 0, 0, ...
13, 2, 1, 0, 1, 1, 0, 0, 0, ...
21, 3, 2, 0, 0, 1, 1, 0, 0, ...
34, 4, 1, 1, 0, 0, 1, 1, 0, ...
55, 5, 1, 2, 0, 0, 0, 1, 1, ...
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MATHEMATICA
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T[n_, k_] := Sum[Binomial[j, n-k*j], {j, 0, Floor[n/k]}]; Table[T[k, n - k + 1], {n, 0, 12}, {k, 0, n}] // Flatten (* Amiram Eldar, Jun 21 2021 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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