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A053495
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Triangle formed by coefficients of numerator polynomials defined by iterating f(u,v) = 1/u - x*v applied to a list of elements {1,2,3,4,...}.
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22
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1, 1, -1, -1, 2, -2, 1, -4, 6, -6, -1, 6, -18, 24, -24, 1, -9, 36, -96, 120, -120, -1, 12, -72, 240, -600, 720, -720, 1, -16, 120, -600, 1800, -4320, 5040, -5040, -1, 20, -200, 1200, -5400, 15120, -35280, 40320, -40320, 1, -25, 300, -2400, 12600
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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FORMULA
| Table[ (-1)^(r+c+1) binomial[Floor[(r+c)/2], Floor[(r-c)/2]] Floor[(r+c+1)/2]! / Floor[(r-c+1)/2]!, {r, 0, 7}, {c, 0, r}]
a[0] := -1; a[1] := 1-x; a[n_] := a[n]= n x a[n-1] + a[n-2] (matches sequence except for a[0]).
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EXAMPLE
| 1, 1 - x, -1 + 2*x - 2*x^2, 1 - 4*x + 6*x^2 - 6*x^3, ...
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MATHEMATICA
| CoefficientList[ #, x ]&/@Numerator[ FoldList[ (1/#1-x#2)&, 1, Range[ 12 ] ]//Together ]
FoldList[(1/#1-x#2)&, 1, Range[4] ]//Together (a simpler version, which shows the rational functions)
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CROSSREFS
| Diagonals give A000142, A001563, A001286, A001809, A001754, A001810, A001755, A001811, A001777. Except for first term, row sums give negative of A058307.
Row sums of positive entries give A001053, those of negative entries give -1*A001040.
Sequence in context: A191490 A061598 A071946 * A096747 A167622 A084606
Adjacent sequences: A053492 A053493 A053494 * A053496 A053497 A053498
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KEYWORD
| sign,tabl,easy,nice
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AUTHOR
| Wouter Meeussen (wouter.meeussen(AT)pandora.be), Jan 27 2001
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