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 A142148 A triangular sequence of polynomial coefficients of an adjusted root product one polynomial set: w(i,n)=If[i == 1, 1/n!, i]; p(x,n)=n!*Product[x - w[i, n], {i, 0, n}]/x. 0
 1, -1, 1, 2, -5, 2, -6, 41, -31, 6, 24, -602, 633, -217, 24, -120, 14554, -18551, 8534, -1681, 120, 720, -519444, 752260, -417755, 111620, -14401, 720, -5040, 25409628, -40466224, 25725825, -8391895, 1486827, -136081, 5040, 40320, -1625771664, 2792773340, -1970053624, 742859705, -162288511 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS Row sums are: {1, 0, -1, 10, -138, 2856, -86280, 3628080, -203207760, 14631281280, -1316818581120}. The one adjusted roots are: Product[w[i,n],{i,1,n}]=1 and sum[Log[w[i,n]],{i,1,n]]=0 so that the first and last coefficients of: Product[x - w[i, n], {i, 0, n}] are one. In this specific case the internal coefficients are skew (not symmetrical). LINKS FORMULA w(i,n)=If[i == 1, 1/n!, i]; p(x,n)=n!*Product[x - w[i, n], {i, 0, n}]/x; t(n,m)=coefficients(p(x,n)). EXAMPLE {1}, {-1, 1}, {2, -5, 2}, {-6, 41, -31, 6}, {24, -602, 633, -217, 24}, {-120, 14554, -18551, 8534, -1681, 120}, {720, -519444, 752260, -417755, 111620, -14401, 720}, {-5040, 25409628, -40466224, 25725825, -8391895, 1486827, -136081, 5040}, {40320, -1625771664, 2792773340, -1970053624, 742859705, -162288511, 20603555, -1411201, 40320}, {-362880, 131682558096, -240842513484, 184707586196, -77901681529, 19831037744, -3129477946, 299738924, -15966721, 362880}, {3628800, -13168196439840, 25401025145736, -20879159852564, 9637237164366, -2762119321689, 511258020084, -61268660466, 4594060854, -195955201, 3628800} MATHEMATICA Clear[w, p]; w[i_, n_] = If[i == 1, 1/n!, i]; p[x_, n_] = n!*Product[x - w[i, n], {i, 0, n}]/x; Table[CoefficientList[p[x, n], x], {n, 0, 10}]; Flatten[%] CROSSREFS Sequence in context: A195621 A267090 A067948 * A142583 A327838 A086956 Adjacent sequences:  A142145 A142146 A142147 * A142149 A142150 A142151 KEYWORD sign,uned AUTHOR Roger L. Bagula and Gary W. Adamson, Sep 15 2008 STATUS approved

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Last modified April 17 03:58 EDT 2021. Contains 343059 sequences. (Running on oeis4.)