OFFSET
0,2
COMMENTS
Signs are important to distinguish this from (for example) A098579.
Sqrt(A(x)) = 1 + 2*x + 2*x^2 - 20*x^3 - 42*x^4 + 572*x^5 ... defines another (new) integer sequence.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
G.f. A(x) satisfies: A(x)^4 = 1 + 16*x*A(x)^2.
A(x) = sqrt(G(4*x)), where G(x) is the g.f. of A182122.
A(x) * A(-x) = 1.
A(x) = sqrt(1 + 8*x + 32*x^2*C(-16*x^2)), where C(x) is the g.f. of A000108.
a(n) = A002420(n)*2^n*(-1)^(n*(n+1)/2). - M. F. Hasler, Aug 14 2015
Conjecture D-finite with recurrence: n*(n-1)*a(n) +(n-1)*(n-2)*a(n-1) +16*(2*n-3)*(2*n-5)*a(n-2) +16*(2*n-5)*(2*n-7)*a(n-3)=0. - R. J. Mathar, Jun 07 2016
EXAMPLE
A(x) = 1 + 4*x + 8*x^2 - 32*x^3 - 160*x^4 + 896*x^5 + 5376*x^6 ...
A(x)^2 = 1 + 8*x + 32*x^2 + 0*x^3 - 512*x^4 + 0*x^5 +16384*x^6 ...
A(x)^4 = 1 + 16*x + 128*x^2 + 512*x^3 + 0*x^4 -8192*x^5 + 0*x^6 ...
MATHEMATICA
CoefficientList[Series[Sqrt[8 x + Sqrt[1 + 64 x^2]], {x, 0, 45}], x] (* Vincenzo Librandi, Aug 12 2015 *)
PROG
(PARI) Vec(sqrt(8*x + sqrt(1 + 64*x^2))) \\ M. F. Hasler, Aug 14 2015
(Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!(Sqrt(8*x+Sqrt(1+64*x^2)))); // G. C. Greubel, Aug 12 2018
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Werner Schulte, Aug 11 2015
STATUS
approved