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A163771
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Triangle interpolating the swinging factorial (A056040) restricted to even indices with its binomial inverse. Same as interpolating the central trinomial coefficients (A002426) with the central binomial coefficients (A000984).
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4
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1, 1, 2, 3, 4, 6, 7, 10, 14, 20, 19, 26, 36, 50, 70, 51, 70, 96, 132, 182, 252, 141, 192, 262, 358, 490, 672, 924, 393, 534, 726, 988, 1346, 1836, 2508, 3432, 1107, 1500, 2034, 2760, 3748, 5094, 6930, 9438, 12870
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| 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)$
where i$ denotes the swinging factorial of i (A056040).
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REFERENCES
| Peter Luschny, "Divide, swing and conquer the factorial and the lcm{1,2,...,n}", preprint, April 2008.
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LINKS
| M. Z. Spivey and L. L. Steil, , The k-Binomial Transforms and the Hankel Transform, J. Integ. Seqs. Vol. 9 (2006), #06.1.1.
Peter Luschny, Swinging Factorial.
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EXAMPLE
| 1
1, 2
3, 4, 6
7, 10, 14, 20
19, 26, 36, 50, 70
51, 70, 96, 132, 182, 252
141, 192, 262, 358, 490, 672, 924
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MAPLE
| For the functions 'DiffTria' and 'swing' see A163770. Computes n rows of the triangle.
a := n -> DiffTria(k->swing(2*k), n, true);
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CROSSREFS
| Sum rows are A163774. Cf. A056040, A163650, A163771, A163772, A002426, A000984.
Sequence in context: A035941 A039854 A032480 * A194855 A117851 A050679
Adjacent sequences: A163768 A163769 A163770 * A163772 A163773 A163774
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KEYWORD
| nonn,tabl
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AUTHOR
| Peter Luschny (peter(AT)luschny.de), Aug 05 2009
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