A053495 - OEIS

%I #7 Mar 30 2012 18:37:42

%S 1,1,-1,-1,2,-2,1,-4,6,-6,-1,6,-18,24,-24,1,-9,36,-96,120,-120,-1,12,

%T -72,240,-600,720,-720,1,-16,120,-600,1800,-4320,5040,-5040,-1,20,

%U -200,1200,-5400,15120,-35280,40320,-40320,1,-25,300,-2400,12600

%N 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,...}.

%F 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}]

%F 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]).

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

%t CoefficientList[ #, x ]&/@Numerator[ FoldList[ (1/#1-x#2)&, 1, Range[ 12 ] ]//Together ]

%t FoldList[(1/#1-x#2)&, 1, Range[4] ]//Together (a simpler version, which shows the rational functions)

%Y Diagonals give A000142, A001563, A001286, A001809, A001754, A001810, A001755, A001811, A001777. Except for first term, row sums give negative of A058307.

%Y Row sums of positive entries give A001053, those of negative entries give -1*A001040.

%K sign,tabl,easy,nice

%O 0,5

%A _Wouter Meeussen_, Jan 27 2001