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A123361 Triangle read by rows: T(n,k) = coefficient of x^k in the polynomial p[n,x] defined by p[0,x]=1, p[1,x]=1+x and p[n,x]=(1+x)(2-x)(3-x)...(n-x) for n >= 2 (0 <= k <= n). 2
1, 1, 1, 2, 1, -1, 6, 1, -4, 1, 24, -2, -17, 8, -1, 120, -34, -83, 57, -13, 1, 720, -324, -464, 425, -135, 19, -1, 5040, -2988, -2924, 3439, -1370, 268, -26, 1, 40320, -28944, -20404, 30436, -14399, 3514, -476, 34, -1, 362880, -300816, -154692, 294328, -160027, 46025, -7798, 782, -43, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
Changing the initial conditions in the recursion produces a different triangular sequence. The result here is a variation of Stirling's numbers of the first kind. The Chang and Sederberg version of this recursion produces an even function in sections.
REFERENCES
Chang and Sederberg, Over and Over Again, MAA, 1997, page 209 (Moving Averages).
LINKS
EXAMPLE
Triangle begins:
1;
1, 1;
2, 1, -1;
6, 1, -4, 1;
24, -2, -17, 8, -1;
120, -34, -83, 57, -13, 1;
720, -324, -464, 425, -135, 19, -1;
5040, -2988, -2924, 3439, -1370, 268, -26, 1;
MAPLE
p[0]:=1: p[1]:=1+x: for n from 2 to 10 do p[n]:=sort(expand((n-x)*p[n-1])) od: for n from 0 to 10 do seq(coeff(p[n], x, k), k=0..n) od; # yields sequence in triangular form
MATHEMATICA
p[ -1, x] = 1; p[0, x] = x + 1; p[k_, x_] := p[k, x] = (-x + k + 1)*p[k - 1, x] w = Table[CoefficientList[p[n, x], x], {n, -1, 10}]; Flatten[w]
CROSSREFS
Cf. A008275.
Sequence in context: A181538 A322128 A125731 * A265315 A179380 A107106
KEYWORD
sign,tabl
AUTHOR
Roger L. Bagula, Nov 09 2006
EXTENSIONS
Edited by N. J. A. Sloane, Nov 24 2006, Jun 17 2007
STATUS
approved

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Last modified April 25 10:22 EDT 2024. Contains 371967 sequences. (Running on oeis4.)