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

%I A049061 #33 Jun 24 2017 00:23:26
%S A049061 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,
%T A049061 1,7,-27,19,19,-27,7,1,1,22,-32,-86,190,-86,-32,22,1,1,21,-54,-54,276,
%U A049061 -276,54,54,-21,-1,1,52,27,-648,1002,0,-1002,648,-27,-52,-1,1,51,-25
%N A049061 Triangle a(n,k) (1<=k<=n) of "signed Eulerian numbers" read by rows.
%C A049061 Identical to rows n=2..n of D(n,k) on page 2 of Tanimoto reference. - _Jonathan Vos Post_, Dec 11 2006
%H A049061 Reinhard Zumkeller, <a href="/A049061/b049061.txt">Rows n = 1..120 of triangle, flattened</a>
%H A049061 J. Desarmenien and D. Foata, <a href="http://www-irma.u-strasbg.fr/~foata/paper/pub62.html">The signed Eulerian Numbers</a>
%H A049061 J. Desarmenien and D. Foata, <a href="http://dx.doi.org/10.1016/0012-365X(92)90364-L">The signed Eulerian numbers</a>, Discrete Math. 99 (1992), no. 1-3, 49-58.
%H A049061 Shinji Tanimoto, <a href="http://arXiv.org/abs/math/0612135">Parity-Alternate Permutations and Signed Eulerian Numbers</a>, arXiv:math/0612135 [math.CO], 2006.
%H A049061 S. Tanimoto, <a href="http://arXiv.org/abs/math.CO/0602263">A new approach to signed Eulerian numbers</a>, arXiv:math/0602263 [math.CO], 2006.
%F A049061 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).
%e A049061 Triangle begins:
%e A049061   1;
%e A049061   1, -1;
%e A049061   1,  0, -1;
%e A049061   1, -1, -1,  1;
%e A049061   1,  2, -6,  2,  1;
%e A049061   ...
%t A049061 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 *)
%o A049061 (Haskell)
%o A049061 a049061 n k = a049061_tabl !! (n-1) !! (k-1)
%o A049061 a049061_row n = a049061_tabl !! (n-1)
%o A049061 a049061_tabl = map fst $ iterate t ([1], 1) where
%o A049061    t (row, k) = (if odd k then us else vs, k + 1) where
%o A049061      us = zipWith (-) (row ++ [0]) ([0] ++ row)
%o A049061      vs = zipWith (+) ((zipWith (*) ks row) ++ [0])
%o A049061                       ([0] ++ (zipWith (*) (reverse ks) row))
%o A049061           where ks = [1..k]
%o A049061 -- _Reinhard Zumkeller_, Jan 30 2013
%Y A049061 Cf. A008292.
%Y A049061 Cf. A080856.
%K A049061 sign,easy,tabl,nice
%O A049061 1,12
%A A049061 _N. J. A. Sloane_
%E A049061 More terms from Larry Reeves (larryr(AT)acm.org), Apr 12 2000
%E A049061 ArXiv URL replaced by its non-cached version - _R. J. Mathar_, Oct 23 2009