OFFSET
0,2
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: A(x) = B(x)^2*(G(x) - 1)/x^2 = B(x)^2*(B(x) - 1)/(x+x^2 - x^2*B(x)), where B(x) is g.f. of A119370 and G(x) is g.f. of A119371 (central terms of A119369).
G.f.: 2*(1-2*x-x^2 -f(x))/( x^2*(1+2*x^3+x^4 +(1+x)^2*f(x))*(1+x^2 +f(x)) where f(x) = sqrt(1-4*x-2*x^2+x^4). - G. C. Greubel, Mar 17 2021
MATHEMATICA
f[x_]:= Sqrt[1-4*x-2*x^2+x^4];
CoefficientList[Series[2*(1-2*x-x^2 -f[x])/(x^2*(1+2*x^3+x^4 +(1+x)^2*f[x])*(1+x^2 +f[x])), {x, 0, 30}], x] (* G. C. Greubel, Mar 17 2021 *)
PROG
(PARI) {a(n)=polcoeff(4/((1+x^2)+sqrt((1+x^2)^2-4*x*(1+x)+x^3*O(x^n)))^2* (2*(1+x)/(1+4*x+x^2 + sqrt((1+4*x+x^2)^2-4*x*(1+x)*(3+2*x)+x^3*O(x^n)))-1)/x^2, n)}
(Sage)
def f(x): return sqrt(1-4*x-2*x^2+x^4)
def A119376_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( 2*(1-2*x-x^2 -f(x))/( x^2*(1+2*x^3+x^4 +(1+x)^2*f(x))*(1+x^2 +f(x)) ) ).list()
A119376_list(30) # G. C. Greubel, Mar 17 2021
(Magma)
R<x>:=PowerSeriesRing(Rationals(), 30);
f:= func< x | Sqrt(1-4*x-2*x^2+x^4) >;
Coefficients(R!( 2*(1-2*x-x^2 -f(x))/( x^2*(1+2*x^3+x^4 +(1+x)^2*f(x))*(1+x^2 +f(x)) ) )); // G. C. Greubel, Mar 17 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, May 17 2006
STATUS
approved