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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
 1, 1, 1, 1, 5, 1, 1, 17, 17, 1, 1, 45, 106, 45, 1, 1, 105, 524, 524, 105, 1, 1, 229, 2231, 4258, 2231, 229, 1, 1, 481, 8547, 28771, 28771, 8547, 481, 1, 1, 989, 30424, 171283, 290126, 171283, 30424, 989, 1, 1, 2009, 102926, 928070, 2505074, 2505074, 928070 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Row sums are: {1, 2, 7, 36, 198, 1260, 9180, 75600, 695520, 7076160, 78926400,...}. The m=0 of the general sequence is A008518. LINKS FORMULA 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)]. EXAMPLE {1}, {1, 1}, {1, 5, 1}, {1, 17, 17, 1}, {1, 45, 106, 45, 1}, {1, 105, 524, 524, 105, 1}, {1, 229, 2231, 4258, 2231, 229, 1}, {1, 481, 8547, 28771, 28771, 8547, 481, 1}, {1, 989, 30424, 171283, 290126, 171283, 30424, 989, 1}, {1, 2009, 102926, 928070, 2505074, 2505074, 928070, 102926, 2009, 1}, {1, 4053, 336109, 4684096, 19330402, 30217078, 19330402, 4684096, 336109, 4053, 1} MATHEMATICA Clear[t, n, k, m]; 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]; Table[t[n, k, m], {m, 0, 10}, {n, 0, 10}, {k, 0, n}]; Table[Flatten[Table[Table[t[n, k, m], {k, 0, n}], {n, 0, 10}]], {m, 0, 10}] Table[Table[Sum[t[n, k, m], {k, 0, n}], {n, 0, 10}], {m, 0, 10}]; CROSSREFS Sequence in context: A174159 A074060 A157637 * A029847 A154334 A178346 Adjacent sequences:  A157178 A157179 A157180 * A157182 A157183 A157184 KEYWORD nonn,tabl AUTHOR Roger L. Bagula and Gary W. Adamson, Feb 24 2009 STATUS approved

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