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A123319 Triangle read by rows: coefficients of polynomials p(k) = (-x + k + 1)*p(k-1), starting p(0)=1, p(1)=1-x.

%I A123319 #20 Mar 12 2014 16:37:01
%S A123319 1,1,-1,3,-4,1,12,-19,8,-1,60,-107,59,-13,1,360,-702,461,-137,19,-1,
%T A123319 2520,-5274,3929,-1420,270,-26,1,20160,-44712,36706,-15289,3580,-478,
%U A123319 34,-1,181440,-422568,375066,-174307,47509,-7882,784,-43,1,1814400,-4407120,4173228,-2118136,649397,-126329,15722
%N A123319 Triangle read by rows: coefficients of polynomials p(k) = (-x + k + 1)*p(k-1), starting p(0)=1, p(1)=1-x.
%C A123319 Recursive polynomial for A008275 shifted up one value of k.
%C A123319 Shifting initial condition in a recurvise polynomial without changing also the function of the iteration variable k produces a new triangular sequence. The result here is a variation of Stirling's numbers of the first kind (A008275). The Chang and Sederberg version of this recursion produces an even function in sections.
%C A123319 Row sums are 0.
%D A123319 Over and Over Again, Chang and Sederberg, MAA, 1997, page 209 (Moving Averages).
%e A123319 Triangle starts:
%e A123319 {1},
%e A123319 {1, -1},
%e A123319 {3, -4, 1},
%e A123319 {12, -19, 8, -1},
%e A123319 {60, -107, 59, -13, 1},
%e A123319 {360, -702, 461, -137, 19, -1},
%e A123319 {2520, -5274, 3929, -1420, 270, -26, 1}
%t A123319 p[0, x] = 1; p[1, x] = -x + 1; p[k_, x_] := p[k, x] = (-x + k + 1)*p[k - 1, x]; w = Table[CoefficientList[p[n, x], x], {n, 0, 10}]; Flatten[w]
%o A123319 (PARI) p(k)=if(k<1,1,if(k<2,1-x,(-x+k+1)*p(k-1)))
%Y A123319 Cf. A008275.
%K A123319 sign,tabl
%O A123319 0,4
%A A123319 _Roger L. Bagula_, Nov 09 2006
%E A123319 Offset corrected to 0. - Wolfdieter Lang, Oct 25 2011.
%E A123319 Edited by _Ralf Stephan_, Sep 08 2013