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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.
4
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).
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
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