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A049061
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Triangle a(n,k) (1<=k<=n) of "signed Eulerian numbers" read by rows.
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7
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1, 1, -1, 1, 0, -1, 1, -1, -1, 1, 1, 2, -6, 2, 1, 1, 1, -8, 8, -1, -1, 1, 8, -19, 0, 19, -8, -1, 1, 7, -27, 19, 19, -27, 7, 1, 1, 22, -32, -86, 190, -86, -32, 22, 1, 1, 21, -54, -54, 276, -276, 54, 54, -21, -1, 1, 52, 27, -648, 1002, 0, -1002, 648, -27, -52, -1, 1, 51, -25
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OFFSET
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1,12
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COMMENTS
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Identical to rows n=2..n of D(n,k) on page 2 of Tanimoto reference. - Jonathan Vos Post, Dec 11 2006
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LINKS
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FORMULA
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a(2n, k) = a(2n-1, k) - a(2n-1, k-1), a(2n+1, k) = k*a(2n, k) + (2n-k+2)*a(2n, k-1).
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EXAMPLE
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Triangle begins:
1;
1, -1;
1, 0, -1;
1, -1, -1, 1;
1, 2, -6, 2, 1;
...
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MATHEMATICA
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a[n_ /; EvenQ[n] && n > 0, k_] := a[n, k] = a[n - 1, k] - a[n - 1, k - 1]; a[n_ /; OddQ[n] && n > 0, k_] := a[n, k] = k*a[n - 1, k] + (n - k + 1)*a[n - 1, k - 1]; a[0, _]=0; a[1, 1]=1; Flatten[Table[a[n, k], {n, 12}, {k, n}]] (* Jean-François Alcover, May 02 2011 *)
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PROG
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(Haskell)
a049061 n k = a049061_tabl !! (n-1) !! (k-1)
a049061_row n = a049061_tabl !! (n-1)
a049061_tabl = map fst $ iterate t ([1], 1) where
t (row, k) = (if odd k then us else vs, k + 1) where
us = zipWith (-) (row ++ [0]) ([0] ++ row)
vs = zipWith (+) ((zipWith (*) ks row) ++ [0])
([0] ++ (zipWith (*) (reverse ks) row))
where ks = [1..k]
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Apr 12 2000
ArXiv URL replaced by its non-cached version - R. J. Mathar, Oct 23 2009
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STATUS
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approved
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