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A261196 Expansion of sqrt(8*x + sqrt(1 + 64*x^2)).

%I A261196 #30 Sep 08 2022 08:46:13
%S A261196 1,4,8,-32,-160,896,5376,-33792,-219648,1464320,9957376,-68796416,
%T A261196 -481574912,3408068608,24343347200,-175272099840,-1270722723840,
%U A261196 9268801044480,67971207659520,-500840477491200,-3706219533434880,27531916534087680,205237923254108160
%N A261196 Expansion of sqrt(8*x + sqrt(1 + 64*x^2)).
%C A261196 Signs are important to distinguish this from (for example) A098579.
%C A261196 Sqrt(A(x)) = 1 + 2*x + 2*x^2 - 20*x^3 - 42*x^4 + 572*x^5 ... defines another (new) integer sequence.
%H A261196 G. C. Greubel, <a href="/A261196/b261196.txt">Table of n, a(n) for n = 0..1000</a>
%F A261196 G.f. A(x) satisfies: A(x)^4 = 1 + 16*x*A(x)^2.
%F A261196 A(x) = sqrt(G(4*x)), where G(x) is the g.f. of A182122.
%F A261196 A(x) * A(-x) = 1.
%F A261196 A(x) = sqrt(1 + 8*x + 32*x^2*C(-16*x^2)), where C(x) is the g.f. of A000108.
%F A261196 a(n) = A002420(n)*2^n*(-1)^(n*(n+1)/2). - _M. F. Hasler_, Aug 14 2015
%F A261196 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
%e A261196 A(x) = 1 + 4*x + 8*x^2 - 32*x^3 - 160*x^4 + 896*x^5 + 5376*x^6 ...
%e A261196 A(x)^2 = 1 + 8*x + 32*x^2 + 0*x^3 - 512*x^4 + 0*x^5 +16384*x^6 ...
%e A261196 A(x)^4 = 1 + 16*x + 128*x^2 + 512*x^3 + 0*x^4 -8192*x^5 + 0*x^6 ...
%t A261196 CoefficientList[Series[Sqrt[8 x + Sqrt[1 + 64 x^2]], {x, 0, 45}], x] (* _Vincenzo Librandi_, Aug 12 2015 *)
%o A261196 (PARI) Vec(sqrt(8*x + sqrt(1 + 64*x^2))) \\ _M. F. Hasler_, Aug 14 2015
%o A261196 (Magma) m:=50; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!(Sqrt(8*x+Sqrt(1+64*x^2)))); // _G. C. Greubel_, Aug 12 2018
%Y A261196 Cf. A000108, A002420, A098579, A182122.
%K A261196 sign,easy
%O A261196 0,2
%A A261196 _Werner Schulte_, Aug 11 2015