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A307394
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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-x)^(k-4))/((1-x)^k+x^k).
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5
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1, 1, 3, 1, 4, 6, 1, 4, 9, 10, 1, 4, 10, 14, 15, 1, 4, 10, 19, 15, 21, 1, 4, 10, 20, 28, 8, 28, 1, 4, 10, 20, 34, 28, -7, 36, 1, 4, 10, 20, 35, 48, 1, -22, 45, 1, 4, 10, 20, 35, 55, 48, -80, -21, 55, 1, 4, 10, 20, 35, 56, 75, 0, -242, 12, 66, 1, 4, 10, 20, 35, 56, 83, 75, -164, -485, 77, 78
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OFFSET
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0,3
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LINKS
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FORMULA
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A(n,k) = Sum_{j=0..floor(n/k)} (-1)^j * binomial(n+3,k*j+3).
A(n,2*k) = Sum_{i=0..n} Sum_{j=0..n-i} (-1)^j * binomial(i+1,k*j+1) * binomial(n-i+1,k*j+1).
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EXAMPLE
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Square array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
3, 4, 4, 4, 4, 4, 4, 4, 4, ...
6, 9, 10, 10, 10, 10, 10, 10, 10, ...
10, 14, 19, 20, 20, 20, 20, 20, 20, ...
15, 15, 28, 34, 35, 35, 35, 35, 35, ...
21, 8, 28, 48, 55, 56, 56, 56, 56, ...
28, -7, 1, 48, 75, 83, 84, 84, 84, ...
36, -22, -80, 0, 75, 110, 119, 120, 120, ...
45, -21, -242, -164, 0, 110, 154, 164, 165, ...
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MATHEMATICA
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T[n_, k_] := Sum[(-1)^j * Binomial[n+3, k*j + 3], {j, 0, Floor[n/k]}]; Table[T[n - k, k], {n, 0, 12}, {k, n, 1, -1}] // Flatten (* Amiram Eldar, May 20 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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