A098580 Expansion of (sqrt(1-8*x)-4*x)/sqrt(1-8*x).

1, -4, -16, -96, -640, -4480, -32256, -236544, -1757184, -13178880, -99573760, -756760576, -5778898944, -44304891904, -340806860800, -2629081497600, -20331563581440, -157569617756160, -1223481737871360, -9515969072332800, -74124390668697600, -578170247215841280

Offset

0,2

Links

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

Formula

G.f.: 1-4*x/sqrt(1-8*x).

a(0)=1, a(n) = -2^(n+1)*(2*(n-1))!/(n-1)!^2.

D-finite with recurrence: (n-1)*a(n) + 4*(3-2*n)*a(n-1) = 0. - R. J. Mathar, Sep 26 2012

From Amiram Eldar, Feb 12 2026: (Start)

Sum_{n>=0} 1/a(n) = 5/7 - 2*arcsin(1/sqrt(8))/(7*sqrt(7)).

Sum_{n>=0} (-1)^n/a(n) = (33 - 2*arcsinh(1/sqrt(8)))/27. (End)

Mathematica

CoefficientList[Series[(Sqrt[1-8x]-4x)/Sqrt[1-8x], {x, 0, 20}], x] (* Harvey P. Dale, May 06 2017 *)

Prog

(PARI) my(x='x+O('x^30)); Vec((sqrt(1-8*x)-4*x)/sqrt(1-8*x)) \\ G. C. Greubel, Feb 03 2018
(Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!((Sqrt(1-8*x)-4*x)/Sqrt(1-8*x))); // G. C. Greubel, Feb 03 2018

Crossrefs

Cf. A059304.

Sequence in context: A263565 A131928 A229138 * A317998 A027745 A293143

Adjacent sequences: A098577 A098578 A098579 * A098581 A098582 A098583

Keyword

easy,sign

Author

Paul Barry, Sep 16 2004

Status

approved