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 A157171 A new general triangle sequence based on the binomial form in three parts:m=3; 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]. 0
 1, 1, 1, 1, 11, 1, 1, 21, 21, 1, 1, 31, 66, 31, 1, 1, 41, 136, 136, 41, 1, 1, 51, 231, 362, 231, 51, 1, 1, 61, 351, 755, 755, 351, 61, 1, 1, 71, 496, 1361, 1870, 1361, 496, 71, 1, 1, 81, 666, 2226, 3906, 3906, 2226, 666, 81, 1, 1, 91, 861, 3396, 7266, 9282, 7266, 3396, 861 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,5 COMMENTS Row sums are: {1, 2, 13, 44, 130, 356, 928, 2336, 5728, 13760, 32512,...}. LINKS FORMULA m=3; 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]. EXAMPLE {1}, {1, 1}, {1, 11, 1}, {1, 21, 21, 1}, {1, 31, 66, 31, 1}, {1, 41, 136, 136, 41, 1}, {1, 51, 231, 362, 231, 51, 1}, {1, 61, 351, 755, 755, 351, 61, 1}, {1, 71, 496, 1361, 1870, 1361, 496, 71, 1}, {1, 81, 666, 2226, 3906, 3906, 2226, 666, 81, 1}, {1, 91, 861, 3396, 7266, 9282, 7266, 3396, 861, 91, 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: A010192 A214326 A105769 * A143685 A168647 A202767 Adjacent sequences:  A157168 A157169 A157170 * A157172 A157173 A157174 KEYWORD nonn,tabl,uned AUTHOR Roger L. Bagula, Feb 24 2009 STATUS approved

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Last modified October 23 23:51 EDT 2019. Contains 328379 sequences. (Running on oeis4.)