cp's OEIS Frontend

A098579 Expansion of sqrt(1-8*x).

%I A098579 #25 Sep 01 2025 05:13:32
%S A098579 1,-4,-8,-32,-160,-896,-5376,-33792,-219648,-1464320,-9957376,
%T A098579 -68796416,-481574912,-3408068608,-24343347200,-175272099840,
%U A098579 -1270722723840,-9268801044480,-67971207659520,-500840477491200,-3706219533434880,-27531916534087680,-205237923254108160
%N A098579 Expansion of sqrt(1-8*x).
%C A098579 If A(x) is the g.f. and B(x) := A(-x), then A(x) * B(x) = A(8*x^2), B(x)^2 - A(x)^2 = 16*x, B(x)^2 + A(x)^2 = 2. Also, if U = x*A(x^2/4), T = x*A(-x^2/4), then (T^2 + U^2)^2 = T^2 - U^2 is a parametrization of the lemniscate of Bernoulli. - _Michael Somos_, Aug 22 2019
%H A098579 G. C. Greubel, <a href="/A098579/b098579.txt">Table of n, a(n) for n = 0..1000</a>
%F A098579 G.f.: 4*x*C(2*x)-1, where C(x) is the g.f. for the Catalan numbers A000108; a(0)=1, a(n)=-2^(n+1)*binomial(2(n-1), n-1)/n, n>0.
%F A098579 D-finite with recurrence: n*a(n) +4*(3-2*n)*a(n-1)=0. - _R. J. Mathar_, Nov 09 2012
%F A098579 0 = a(n)*(+64*a(n+1) -20*a(n+2)) +a(n+1)*(+4*a(n+1) +a(n+2)) for all n in Z. - _Michael Somos_, Aug 22 2019
%F A098579 a(n) ~ -2^(3*n-1) / (sqrt(Pi) * n^(3/2)). - _Amiram Eldar_, Sep 01 2025
%e A098579 G.f. = 1 - 4*x - 8*x^2 - 32*x^3 - 160*x^4 - 896*x^5 - 5376*x^6 + ... - _Michael Somos_, Aug 22 2019
%t A098579 CoefficientList[Series[Sqrt[1-8x],{x,0,30}],x] (* or *) Table[(8^(x-1) Pochhammer[-(1/2),x-1])/Pochhammer[1,x-1],{x,30}] (* _Harvey P. Dale_, Jan 24 2015 *)
%t A098579 a[ n_] := If[ n < 1, Boole[n == 0], -CatalanNumber[n - 1] 2^(n + 1)]; (* _Michael Somos_, Aug 22 2019 *)
%o A098579 (PARI) my(x='x+O('x^30)); Vec(sqrt(1-8*x)) \\ _G. C. Greubel_, Feb 03 2018
%o A098579 (Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!(Sqrt(1-8*x))); // _G. C. Greubel_, Feb 03 2018
%Y A098579 Cf. A052701, A002420.
%K A098579 easy,sign,changed
%O A098579 0,2
%A A098579 _Paul Barry_, Sep 16 2004