A049061 Triangle a(n,k) (1<=k<=n) of "signed Eulerian numbers" read by rows.

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

Offset

1,12

Comments

Identical to rows n=2..n of D(n,k) on page 2 of Tanimoto reference. - Jonathan Vos Post, Dec 11 2006

Links

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened

J. Desarmenien and D. Foata, The signed Eulerian Numbers

J. Desarmenien and D. Foata, The signed Eulerian numbers, Discrete Math. 99 (1992), no. 1-3, 49-58.

Shinji Tanimoto, Parity-Alternate Permutations and Signed Eulerian Numbers, arXiv:math/0612135 [math.CO], 2006.

S. Tanimoto, A new approach to signed Eulerian numbers, arXiv:math/0602263 [math.CO], 2006.

Formula

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).

Example

Triangle begins:
  1;
  1, -1;
  1,  0, -1;
  1, -1, -1,  1;
  1,  2, -6,  2,  1;
  ...

Mathematica

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 *)

Prog

(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]
-- Reinhard Zumkeller, Jan 30 2013

Crossrefs

Cf. A008292.

Cf. A080856.

Sequence in context: A363737 A004544 A010590 * A342670 A298782 A082516

Adjacent sequences: A049058 A049059 A049060 * A049062 A049063 A049064

Keyword

sign,easy,tabl,nice

Author

N. J. A. Sloane

Extensions

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

Status

approved