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).
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
KEYWORD
sign,uned
AUTHOR
Roger L. Bagula and Gary W. Adamson, Sep 15 2008
STATUS
approved