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A157178 A new general triangle sequence based on the Eulerian form in three parts:m=2; 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 A157178 #2 Oct 12 2012 14:54:56
%S A157178 1,1,1,1,8,1,1,21,21,1,1,46,142,46,1,1,95,644,644,95,1,1,192,2439,
%T A157178 5416,2439,192,1,1,385,8415,34879,34879,8415,385,1,1,770,27556,192286,
%U A157178 358454,192286,27556,770,1,1,1539,87486,962090,3001044,3001044,962090,87486
%N A157178 A new general triangle sequence based on the Eulerian form in three parts:m=2; 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 A157178 Row sums are:
%C A157178 {1, 2, 10, 44, 236, 1480, 10680, 87360, 799680, 8104320, 90115200,...}.
%C A157178 The m=1 of the general sequence is A008518.
%F A157178 m=2;
%F A157178 t0(n,k)=If[n*k == 0, 1, Sum[(-1)^j Binomial[n + 1, j](k + 1 - j)^n, {j, 0, k + 1}]];
%F A157178 t(n,k,m)=If[n == 0, 1, ( m*(n - k) + 1)*t0(n - 1 + 1, k - 1) +
%F A157178 (m*k + 1)*t0(n - 1 + 1, k) +
%F A157178 m*k*(n - k)*t0(n - 2 + 1, k - 1)].
%e A157178 {1},
%e A157178 {1, 1},
%e A157178 {1, 8, 1},
%e A157178 {1, 21, 21, 1},
%e A157178 {1, 46, 142, 46, 1},
%e A157178 {1, 95, 644, 644, 95, 1},
%e A157178 {1, 192, 2439, 5416, 2439, 192, 1},
%e A157178 {1, 385, 8415, 34879, 34879, 8415, 385, 1},
%e A157178 {1, 770, 27556, 192286, 358454, 192286, 27556, 770, 1},
%e A157178 {1, 1539, 87486, 962090, 3001044, 3001044, 962090, 87486, 1539, 1},
%e A157178 {1, 3076, 272485, 4517480, 21945914, 36637288, 21945914, 4517480, 272485, 3076, 1}
%t A157178 Clear[t, n, k, m];
%t A157178 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 A157178 Table[t[n, k, m], {m, 0, 10}, {n, 0, 10}, {k, 0, n}];
%t A157178 Table[Flatten[Table[Table[t[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 10}]
%t A157178 Table[Table[Sum[t[n, k, m], {k, 0, n}], {n, 0, 10}], {m, 0, 10}];
%Y A157178 A008518
%K A157178 nonn,tabl
%O A157178 0,5
%A A157178 _Roger L. Bagula_ and _Gary W. Adamson_, Feb 24 2009