A053495 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,...}.

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

Offset: 0

Author

Wouter Meeussen, Jan 27 2001

Keywords

• sign
• tabl
• easy
• nice

Examples

			1, 1 - x, -1 + 2*x - 2*x^2, 1 - 4*x + 6*x^2 - 6*x^3, ...
		

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.

Programs

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)

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]).