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

1, 1, -1, 3, -4, 1, 12, -19, 8, -1, 60, -107, 59, -13, 1, 360, -702, 461, -137, 19, -1, 2520, -5274, 3929, -1420, 270, -26, 1, 20160, -44712, 36706, -15289, 3580, -478, 34, -1, 181440, -422568, 375066, -174307, 47509, -7882, 784, -43, 1, 1814400, -4407120, 4173228, -2118136, 649397, -126329, 15722

Offset: 0

Author

Roger L. Bagula, Nov 09 2006

Keywords

• sign
• tabl

Comments

Recursive polynomial for A008275 shifted up one value of k.

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.

Row sums are 0.

Examples

			Triangle starts:
{1},
{1, -1},
{3, -4, 1},
{12, -19, 8, -1},
{60, -107, 59, -13, 1},
{360, -702, 461, -137, 19, -1},
{2520, -5274, 3929, -1420, 270, -26, 1}
		

References

• Over and Over Again, Chang and Sederberg, MAA, 1997, page 209 (Moving Averages).

Crossrefs

Cf. A008275.

Programs

Mathematica

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]

PARI

p(k)=if(k<1,1,if(k<2,1-x,(-x+k+1)*p(k-1)))

Extensions

Offset corrected to 0. - Wolfdieter Lang, Oct 25 2011.

Edited by Ralf Stephan, Sep 08 2013