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A symmetrical triangular sequence adjusted by -t[n,-1]+1 :t(n,m)=If[n == 0, 1, Sum[Binomial[n, k]*Eulerian[n + 1, k]*(-1)^k*(m + 1 - k)^n, {k, 0, n}]]
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%I #2 Mar 30 2012 17:34:39

%S 1,1,1,1,7,1,1,181,181,1,1,751,1761,751,1,1,-323399,-354899,-354899,

%T -323399,1,1,-20873579,-27315455,-27944027,-27315455,-20873579,1,1,

%U 1429118713,1312010281,1219622545,1219622545,1312010281,1429118713,1,1

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

%C Row sums are:

%C 1, 2, 9, 364, 3265, -1356594, -124322093, 7921503080, 3867350839137,

%C 254033305455610, -174275807582741689,...

%F t(n,m)=If[n == 0, 1, Sum[Binomial[n, k]*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, 7, 1},

%e {1, 181, 181, 1},

%e {1, 751, 1761, 751, 1},

%e {1, -323399, -354899, -354899, -323399, 1},

%e {1, -20873579, -27315455, -27944027, -27315455, -20873579, 1},

%e {1, 1429118713, 1312010281, 1219622545, 1219622545, 1312010281, 1429118713, 1},

%e {1, 481723390135, 577653410305, 583205489911, 582186258433, 583205489911, 577653410305, 481723390135, 1},

%e {1, 19475791988401, 33530084432461, 36881727519781, 37129048787161, 37129048787161, 36881727519781, 33530084432461, 19475791988401, 1},

%e {1, -17832181758066323, -19958592533330175, -19793122506695075, -19704919818993023, -19698174348572499, -19704919818993023, -19793122506695075, -19958592533330175, -17832181758066323, 1}

%t << DiscreteMath`Combinatorica`

%t t[n_, m_] = If[n == 0, 1, Sum[Binomial[n, k]*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