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.

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

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

Table of n, a(n) for n=1..42.

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