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

0,4

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.

References

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

Links

Table of n, a(n) for n=0..51.

Example

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}

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]

Prog

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

Crossrefs

Cf. A008275.

Sequence in context: A216154 A325174 A109956 * A076785 A110506 A114189

Adjacent sequences: A123316 A123317 A123318 * A123320 A123321 A123322

Keyword

sign,tabl

Author

Roger L. Bagula, Nov 09 2006

Extensions

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

Edited by Ralf Stephan, Sep 08 2013

Status

approved