A049150 Recip transform of 2*(1 + x^2)-1/(1-x).

1, 1, 1, 1, 3, 15, 59, 187, 533, 1541, 4893, 16797, 58663, 201347, 679767, 2294967, 7850121, 27247369, 95375225, 334643225, 1174649611, 4129971863, 14570334995, 51610458291, 183436895645, 653582527693, 2333035219285, 8342630973365

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

Links

Table of n, a(n) for n=0..27.

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

Crossrefs

Sequence in context: A017949 A263173 A049178 * A309564 A062473 A218200

Adjacent sequences: A049147 A049148 A049149 * A049151 A049152 A049153

Keyword

nonn

Author

Olivier Gérard

Status

approved