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A142148
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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.
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0
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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
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OFFSET
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1,4
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COMMENTS
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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).
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LINKS
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FORMULA
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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)).
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EXAMPLE
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{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}
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MATHEMATICA
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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[%]
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CROSSREFS
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KEYWORD
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sign,uned
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AUTHOR
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STATUS
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approved
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