%I #2 Oct 12 2012 14:54:56

%S 1,1,1,1,3,1,1,9,9,1,1,23,50,23,1,1,53,236,236,53,1,1,115,983,1822,

%T 983,115,1,1,241,3723,11995,11995,3723,241,1,1,495,13168,70369,117534,

%U 70369,13168,495,1,1,1005,44382,377918,997974,997974,377918,44382,1005,1,1

%N A new general triangle sequence based on the Eulerian form in three parts ( subtraction):m=1; t0(n,k)=If[n*k == 0, 1, Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}]] t(n,k,m)=If[n == 0, 1, ( m*(n - k) + 1)*t0(n - 1 + 1, k - 1) + (m*k + 1)*t0(n - 1 + 1, k) - m*k*(n - k)*t0(n - 2 + 1, k - 1)].

%C Row sums are:

%C {1, 2, 5, 20, 98, 580, 4020, 31920, 285600, 2842560, 31147200,...}.

%C The m=0 of the general sequence is A008518.

%F m=1;

%F t0(n,k)=If[n*k == 0, 1, Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}]];

%F t(n,k,m)=If[n == 0, 1, ( m*(n - k) + 1)*t0(n - 1 + 1, k - 1) +

%F (m*k + 1)*t0(n - 1 + 1, k) +

%F m*k*(n - k)*t0(n - 2 + 1, k - 1)].

%e {1},

%e {1, 1},

%e {1, 3, 1},

%e {1, 9, 9, 1},

%e {1, 23, 50, 23, 1},

%e {1, 53, 236, 236, 53, 1},

%e {1, 115, 983, 1822, 983, 115, 1},

%e {1, 241, 3723, 11995, 11995, 3723, 241, 1},

%e {1, 495, 13168, 70369, 117534, 70369, 13168, 495, 1},

%e {1, 1005, 44382, 377918, 997974, 997974, 377918, 44382, 1005, 1},

%e {1, 2027, 144605, 1896720, 7620498, 11819498, 7620498, 1896720, 144605, 2027, 1}

%t Clear[t, n, k, m];

%t t[n_, k_, m_] = (m*(n - k) + 1)*Binomial[n - 1, k - 1] + (m*k + 1)*Binomial[n - 1, k] - m*k*(n - k)*Binomial[n - 2, k - 1];

%t Table[t[n, k, m], {m, 0, 10}, {n, 0, 10}, {k, 0, n}];

%t Table[Flatten[Table[Table[t[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 10}]

%t Table[Table[Sum[t[n, k, m], {k, 0, n}], {n, 0, 10}], {m, 0, 10}];

%Y A008518

%K nonn,tabl

%O 0,5

%A _Roger L. Bagula_ and _Gary W. Adamson_, Feb 24 2009