%I #12 Jul 24 2023 09:21:34

%S 1,1,1,1,3,15,59,187,533,1541,4893,16797,58663,201347,679767,2294967,

%T 7850121,27247369,95375225,334643225,1174649611,4129971863,

%U 14570334995,51610458291,183436895645,653582527693,2333035219285,8342630973365

%N Recip transform of 2*(1 + x^2)-1/(1-x).

%C Sign diagram of generating sequence: +-+------------...

%C 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

%F 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

%p Order := 80 ;

%p recip := proc(gf)

%p local g ;

%p g := x*algsubs(x=-x,gf) ;

%p solve(series(g, x)=y, x) :

%p convert(%, polynom) :

%p seq((-1)^(i+1)*coeff(%, y, i), i=1..Order-1) ;

%p end proc:

%p recip(2*(1+x^2)-1/(1-x)) ; # _R. J. Mathar_, Jul 24 2023

%K nonn

%O 0,5

%A _Olivier Gérard_