

A049150


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


1



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
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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



FORMULA

Dfinite with recurrence +11*n*(n1)*(3451*n 9253)*(n+1)*a(n) n *(n1)*(262463*n^2 830780*n +369555)*a(n1) +2 *(n1)*(369903*n^3 1717166*n^2 +2277590*n 819714)*a(n2) +2*(560881*n^4 +3987159*n^3 9909552*n^2 +9765832*n 2854248)*a(n3) +3*(32895*n^4 244521*n^3 +411699*n^2 +581243*n 1516376)*a(n4) 3*(3*n11)*(17*n+411) *(n4)*(3*n13)*a(n5)=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..Order1) ;
end proc:


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



