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A157177 A new general triangle sequence based on the Eulerian form in three parts: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)].

%I A157177 #2 Oct 12 2012 14:54:56
%S A157177 1,1,1,1,5,1,1,13,13,1,1,29,82,29,1,1,61,368,368,61,1,1,125,1399,3010,
%T A157177 1399,125,1,1,253,4863,19243,19243,4863,253,1,1,509,16048,106099,
%U A157177 194846,106099,16048,509,1,1,1021,51298,532466,1622734,1622734,532466,51298
%N A157177 A new general triangle sequence based on the Eulerian form in three parts: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 A157177 Row sums are:
%C A157177 {1, 2, 7, 28, 142, 860, 6060, 48720, 440160, 4415040, 48686400,...}.
%C A157177 The m=1 of the general sequence is A008518.
%F A157177 m=1;
%F A157177 t0(n,k)=If[n*k == 0, 1, Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}]];
%F A157177 t(n,k,m)=If[n == 0, 1, ( m*(n - k) + 1)*t0(n - 1 + 1, k - 1) +
%F A157177 (m*k + 1)*t0(n - 1 + 1, k) +
%F A157177 m*k*(n - k)*t0(n - 2 + 1, k - 1)].
%e A157177 {1},
%e A157177 {1, 1},
%e A157177 {1, 5, 1},
%e A157177 {1, 13, 13, 1},
%e A157177 {1, 29, 82, 29, 1},
%e A157177 {1, 61, 368, 368, 61, 1},
%e A157177 {1, 125, 1399, 3010, 1399, 125, 1},
%e A157177 {1, 253, 4863, 19243, 19243, 4863, 253, 1},
%e A157177 {1, 509, 16048, 106099, 194846, 106099, 16048, 509, 1},
%e A157177 {1, 1021, 51298, 532466, 1622734, 1622734, 532466, 51298, 1021, 1},
%e A157177 {1, 2045, 160669, 2510256, 11855730, 19628998, 11855730, 2510256, 160669, 2045, 1}
%t A157177 Clear[t, n, k, m];
%t A157177 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 A157177 Table[t[n, k, m], {m, 0, 10}, {n, 0, 10}, {k, 0, n}];
%t A157177 Table[Flatten[Table[Table[t[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 10}]
%t A157177 Table[Table[Sum[t[n, k, m], {k, 0, n}], {n, 0, 10}], {m, 0, 10}];
%Y A157177 A008518
%K A157177 nonn,tabl
%O A157177 0,5
%A A157177 _Roger L. Bagula_ and _Gary W. Adamson_, Feb 24 2009