OFFSET
0,5
COMMENTS
Sign diagram of generating sequence: +-+------------...
In A049150 to A049170 Gerard defines the "recip" transform as a mix of sign reversals, shifts and the series revert transformation: the recip transform of g(x), a rational ordinary function, defines a sequence of +1 and -1, summarized in the comments as a sequence of signs. Then each second of these signs is flipped, equivalent to the substitution x->(-x) in the generating function. The offset is increased by 1, equivalent to multiplication of the generating function by x. The usual series reversion (see the standard definitions) of a power series x+O(x^2) is applied, the result is divided by x (i.e,. the offset is changed from 1 back to 0) and again the sign of each 2nd term is flipped (equivalent to x->-x in the final o.g.f.). - R. J. Mathar, Jul 24 2023
FORMULA
D-finite with recurrence +11*n*(n-1)*(3451*n -9253)*(n+1)*a(n) -n *(n-1)*(262463*n^2 -830780*n +369555)*a(n-1) +2 *(n-1)*(369903*n^3 -1717166*n^2 +2277590*n -819714)*a(n-2) +2*(-560881*n^4 +3987159*n^3 -9909552*n^2 +9765832*n -2854248)*a(n-3) +3*(32895*n^4 -244521*n^3 +411699*n^2 +581243*n -1516376)*a(n-4) -3*(3*n-11)*(17*n+411) *(n-4)*(3*n-13)*a(n-5)=0. - R. J. Mathar, Jul 24 2023
MAPLE
Order := 80 ;
recip := proc(gf)
local g ;
g := x*algsubs(x=-x, gf) ;
solve(series(g, x)=y, x) :
convert(%, polynom) :
seq((-1)^(i+1)*coeff(%, y, i), i=1..Order-1) ;
end proc:
recip(2*(1+x^2)-1/(1-x)) ; # R. J. Mathar, Jul 24 2023
KEYWORD
nonn
AUTHOR
STATUS
approved