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
Examples
Triangle begins: 1; 1, -1; 1, 0, -1; 1, -1, -1, 1; 1, 2, -6, 2, 1; ...
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.
Programs
-
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 -
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 *)
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).
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
Comments