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%I #2 Mar 30 2012 17:34:39
%S 1,2,2,6,3,-5,-2,24,-16,-64,-24,120,-350,-679,-151,74,16,720,-5076,
%T -6746,1198,2692,544,5040,-73332,-55628,80239,68081,8858,-2000,-272,
%U 40320,-1135296,-18712,2522252,1417588,-83312,-155392,-19840,362880
%N A triangular sequence of polynomial coefficients:p(x,n)=Sum[Eulerian[n + 1, k]*Product[x + i, {i, 0, n - k + 1}]*(-x)^k, {k, 0, n}]/x
%C Row sums are:
%C {1, 4, 2, -80, -970, -6668, 30986, 2567608, 67974542, 1084990396, -6362961886,...}
%F p(x,n)=Sum[Eulerian[n + 1, k]*Product[x + i, {i, 0, n - k + 1}]*(-x)^k, {k, 0, n}]/x
%F t(n,m)=coefficients(p(x,n))
%e {1},
%e {2, 2},
%e {6, 3, -5, -2},
%e {24, -16, -64, -24},
%e {120, -350, -679, -151, 74, 16},
%e {720, -5076, -6746, 1198, 2692, 544},
%e {5040, -73332, -55628, 80239, 68081, 8858, -2000, -272},
%e {40320, -1135296, -18712, 2522252, 1417588, -83312, -155392, -19840},
%e {362880, -19214064, 19785852, 68794296, 21423037, -14938145, -7669234, -662752, 84736, 7936},
%e {3628800, -356968800, 901740888, 1768795280, -83612926, -876416978, -291269756, 6024320, 12008192, 1061376},
%e {39916800, -7267692960, 33899921496, 41987822092, -30465270802, -40831787105, -7672559733, 2867304822, 1021675584, 63226000, -5164288, -353792}
%t << DiscreteMath`Combinatorica`
%t Clear[p, n, x, k];
%t p[x, 0] := 1;
%t p[x_, n_] := Sum[Eulerian[n + 1, k]* Product[x + i, {i, 0, n - k + 1}]*(-x)^k, {k, 0, n}]/x;
%t Table[CoefficientList[FullSimplify[ExpandAll[p[x, n]]], x], {n, 0, 10}];
%t Flatten[%]
%Y Cf. A174789
%K sign,tabl,uned
%O 0,2
%A _Roger L. Bagula_, Mar 30 2010
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