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

1, -4, -8, -32, -160, -896, -5376, -33792, -219648, -1464320, -9957376, -68796416, -481574912, -3408068608, -24343347200, -175272099840, -1270722723840, -9268801044480, -67971207659520, -500840477491200, -3706219533434880, -27531916534087680, -205237923254108160

Offset: 0

Author

Paul Barry, Sep 16 2004

Keywords

• easy
• sign
• changed

Comments

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

Examples

			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
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A052701, A002420.

Programs

Magma

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!(Sqrt(1-8*x))); // G. C. Greubel, Feb 03 2018

Mathematica

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 *)
a[ n_] := If[ n < 1, Boole[n == 0], -CatalanNumber[n - 1] 2^(n + 1)]; (* Michael Somos, Aug 22 2019 *)

PARI

my(x='x+O('x^30)); Vec(sqrt(1-8*x)) \\ G. C. Greubel, Feb 03 2018

Formula

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.

D-finite with recurrence: n*a(n) +4*(3-2*n)*a(n-1)=0. - R. J. Mathar, Nov 09 2012

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

a(n) ~ -2^(3*n-1) / (sqrt(Pi) * n^(3/2)). - Amiram Eldar, Sep 01 2025