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Search: id:A123361
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| A123361 |
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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). |
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+0
1
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| 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, 3628800, -3371040
(list; table; graph; listen)
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OFFSET
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0,4
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COMMENT
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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.
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REFERENCES
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Chang and Sederberg, Over and Over Again, MAA, 1997, page 209 (Moving Averages).
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EXAMPLE
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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}
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MAPLE
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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
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MATHEMATICA
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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]
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CROSSREFS
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Cf. A008275.
Sequence in context: A096162 A053383 A125731 this_sequence A107106 A119502 A142156
Adjacent sequences: A123358 A123359 A123360 this_sequence A123362 A123363 A123364
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KEYWORD
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sign,tabl
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AUTHOR
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Roger Bagula (rlbagulatftn(AT)yahoo.com), Nov 09 2006
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EXTENSIONS
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Edited by N. J. A. Sloane (njas(AT)research.att.com), Nov 24 2006, Jun 17 2007
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