A085403 Expansion of (1-x+sqrt(1-6x+x^2))/2 in powers of x.

1, -2, -2, -6, -22, -90, -394, -1806, -8558, -41586, -206098, -1037718, -5293446, -27297738, -142078746, -745387038, -3937603038, -20927156706, -111818026018, -600318853926, -3236724317174, -17518619320890, -95149655201962, -518431875418926, -2832923350929742

Offset: 0

Author

Michael Somos, Jun 28 2003

Keywords

• sign

Comments

Series reversion of x(Sum_{k>=0} a(k)x^k) is x(Sum_{k>=0} A027307(k)x^k).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

A minor variation of A006318. a(n)=-A006318(n-1), n>1.

Programs

Mathematica

CoefficientList[Series[(1-x+Sqrt[1-6x+x^2])/2,{x,0,30}],x] (* Harvey P. Dale, Jun 13 2013 *)

PARI

a(n)=polcoeff((1-x+sqrt(1-6*x+x^2+x*O(x^n)))/2,n)

Formula

G.f.: (1-x+sqrt(1-6x+x^2))/2. (=1/g.f. A006318)

Given g.f. A(x), y=A(x)x satisfies 0=f(x, y) where f(x, y)=y(y-x)+(x+y)x^2 . - Michael Somos, May 23 2005

G.f.: Q(0) where Q(k) = 1 + k*(1-x) - x - x*(k+1)*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Mar 14 2013

a(n) ~ -sqrt(3*sqrt(2)-4) * (3+2*sqrt(2))^n / (2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Apr 20 2014

D-finite with recurrence: n*a(n) +3*(-2*n+3)*a(n-1) +(n-3)*a(n-2)=0. - R. J. Mathar, Jan 20 2020