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A163945
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Triangle interpolating between (-1)^n (A033999) and the swinging factorial function (A056040) restricted to odd indices (2n+1)$ (A002457).
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0
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1, -1, 6, 1, -12, 30, -1, 18, -90, 140, 1, -24, 180, -560, 630, -1, 30, -300, 1400, -3150, 2772, 1, -36, 450, -2800, 9450, -16632, 12012, -1, 42, -630, 4900, -22050, 58212, -84084, 51480, 1, -48, 840, -7840, 44100, -155232, 336336, -411840, 218790
(list; 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) = (-1)^(n-k) binomial(n,k) (2*k+1)$
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
| Peter Luschny, Swinging Factorial.
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EXAMPLE
| 1
-1, 6
1, -12, 30
-1, 18, -90, 140
1, -24, 180, -560, 630
-1, 30, -300, 1400, -3150, 2772
1, -36, 450, -2800, 9450, -16632, 12012
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MAPLE
| swing := proc(n) option remember; if n = 0 then 1 elif
irem(n, 2) = 1 then swing(n-1)*n else 4*swing(n-1)/n fi end:
a := proc(n, k) (-1)^(n-k)*binomial(n, k)*swing(2*k+1) end:
seq(print(seq(a(n, k), k=0..n)), n=0..8);
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CROSSREFS
| Row sums are the inverse binomial transform of the beta numbers (A163872).
Cf. A163649, A098473, A056040.
Sequence in context: A040041 A162933 A090850 * A013613 A122508 A171006
Adjacent sequences: A163942 A163943 A163944 * A163946 A163947 A163948
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KEYWORD
| sign
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AUTHOR
| Peter Luschny (peter(AT)luschny.de), Aug 07 2009
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