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 A157181 A new general triangle sequence based on the Eulerian form in three parts ( subtraction):m=3; 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)]. 0

%I

%S 1,1,1,1,5,1,1,17,17,1,1,45,106,45,1,1,105,524,524,105,1,1,229,2231,

%T 4258,2231,229,1,1,481,8547,28771,28771,8547,481,1,1,989,30424,171283,

%U 290126,171283,30424,989,1,1,2009,102926,928070,2505074,2505074,928070

%N A new general triangle sequence based on the Eulerian form in three parts ( subtraction):m=3; 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, 7, 36, 198, 1260, 9180, 75600, 695520, 7076160, 78926400,...}.

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

%F m=3;

%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, 5, 1},

%e {1, 17, 17, 1},

%e {1, 45, 106, 45, 1},

%e {1, 105, 524, 524, 105, 1},

%e {1, 229, 2231, 4258, 2231, 229, 1},

%e {1, 481, 8547, 28771, 28771, 8547, 481, 1},

%e {1, 989, 30424, 171283, 290126, 171283, 30424, 989, 1},

%e {1, 2009, 102926, 928070, 2505074, 2505074, 928070, 102926, 2009, 1},

%e {1, 4053, 336109, 4684096, 19330402, 30217078, 19330402, 4684096, 336109, 4053, 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

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