%I #2 Mar 30 2012 17:34:39

%S 1,1,1,1,3,1,1,49,49,1,1,553,801,553,1,1,-11999,-9839,-9839,-11999,1,

%T 1,-1125919,-1374463,-1376351,-1374463,-1125919,1,1,-25622015,

%U -41341439,-44792831,-44792831,-41341439,-25622015,1,1,3235358337,3370753537

%N A symmetrical triangular sequence adjusted by -t[n,-1]+1 :t(n,m)=If[n == 0, 1, Sum[Eulerian[n + 1, k]*(-1)^k*(m + 1 - k)^n, {k, 0, n}]]

%C Row sums are:

%C 1, 2, 5, 100, 1909, -43674, -6377113, -223512568, 22735136265, 4265143649290,

%C 148391121390731,...

%F t(n,m)=If[n == 0, 1, Sum[Eulerian[n + 1, k]*(-1)^k*(m + 1 - k)^n, {k, 0, n}]];

%F out_n,m=t(n,m)-t(n,-1)+1

%e {1},

%e {1, 1},

%e {1, 3, 1},

%e {1, 49, 49, 1},

%e {1, 553, 801, 553, 1},

%e {1, -11999, -9839, -9839, -11999, 1},

%e {1, -1125919, -1374463, -1376351, -1374463, -1125919, 1},

%e {1, -25622015, -41341439, -44792831, -44792831, -41341439, -25622015, 1},

%e {1, 3235358337, 3370753537, 3187620481, 3147671553, 3187620481, 3370753537, 3235358337, 1},

%e {1, 437895275521, 556690955521, 569908535041, 568077058561, 568077058561, 569908535041, 556690955521, 437895275521, 1},

%e {1, 7157082826369, 16706096371713, 19892759440513, 20296689312769, 20285865488001, 20296689312769, 19892759440513, 16706096371713, 7157082826369, 1}

%t << DiscreteMath`Combinatorica`

%t t[n_, m_] = If[n == 0, 1, Sum[Eulerian[n + 1, k]*(-1)^k*(m + 1 - k)^n, {k, 0, n}]];

%t Table[Table[t[n, m] - t[n, -1] + 1, {m, -1, n - 1}], {n, 0, 10}];

%t Flatten[%]

%K sign,tabl,uned

%O 0,5

%A _Roger L. Bagula_, Mar 23 2010