login
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

%I #2 Oct 12 2012 14:54:51

%S 1,-1,1,2,-5,2,-6,41,-31,6,24,-602,633,-217,24,-120,14554,-18551,8534,

%T -1681,120,720,-519444,752260,-417755,111620,-14401,720,-5040,

%U 25409628,-40466224,25725825,-8391895,1486827,-136081,5040,40320,-1625771664,2792773340,-1970053624,742859705,-162288511

%N 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.

%C Row sums are:

%C {1, 0, -1, 10, -138, 2856, -86280, 3628080, -203207760, 14631281280, -1316818581120}.

%C The one adjusted roots are:

%C Product[w[i,n],{i,1,n}]=1

%C and

%C sum[Log[w[i,n]],{i,1,n]]=0

%C so that the first and last coefficients of:

%C Product[x - w[i, n], {i, 0, n}]

%C are one. In this specific case the internal coefficients are skew

%C (not symmetrical).

%F 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)).

%e {1},

%e {-1, 1},

%e {2, -5, 2},

%e {-6, 41, -31, 6},

%e {24, -602, 633, -217, 24},

%e {-120, 14554, -18551, 8534, -1681, 120},

%e {720, -519444, 752260, -417755, 111620, -14401, 720},

%e {-5040, 25409628, -40466224, 25725825, -8391895, 1486827, -136081, 5040}, {40320, -1625771664, 2792773340, -1970053624, 742859705, -162288511, 20603555, -1411201, 40320},

%e {-362880, 131682558096, -240842513484, 184707586196, -77901681529, 19831037744, -3129477946, 299738924, -15966721, 362880},

%e {3628800, -13168196439840, 25401025145736, -20879159852564, 9637237164366, -2762119321689, 511258020084, -61268660466, 4594060854, -195955201, 3628800}

%t 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[%]

%K sign,uned

%O 1,4

%A _Roger L. Bagula_ and _Gary W. Adamson_, Sep 15 2008