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A163772
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Triangle interpolating the swinging factorial (A056040) restricted to odd indices with its binomial inverse.
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5
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1, 5, 6, 19, 24, 30, 67, 86, 110, 140, 227, 294, 380, 490, 630, 751, 978, 1272, 1652, 2142, 2772, 2445, 3196, 4174, 5446, 7098, 9240, 12012, 7869, 10314, 13510, 17684, 23130, 30228, 39468, 51480
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OFFSET
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0,2
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COMMENTS
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Triangle read by rows. For n >= 0, k >= 0 let
T(n,k) = Sum_{i=k..n} (-1)^(n-i)*binomial(n-k,n-i)*(2i+1)$ where i$ denotes the swinging factorial of i (A056040).
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LINKS
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EXAMPLE
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1
5, 6
19, 24, 30
67, 86, 110, 140
227, 294, 380, 490, 630
751, 978, 1272, 1652, 2142, 2772
2445, 3196, 4174, 5446, 7098, 9240, 12012
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MAPLE
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For the functions 'DiffTria' and 'swing' see A163770. Computes n rows of the triangle.
a := n -> DiffTria(k->swing(2*k+1), n, true);
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MATHEMATICA
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sf[n_] := n!/Quotient[n, 2]!^2; t[n_, k_] := Sum[ (-1)^(n-i)*Binomial[n-k, n-i]*sf[2*i+1], {i, k, n}]; Table[t[n, k], {n, 0, 7}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 28 2013 *)
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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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