A086990 Expansion of (1+4x-sqrt(1+4x^2))/(4+6x) in powers of x.

0, 1, -2, 3, -4, 6, -10, 15, -20, 30, -52, 78, -96, 144, -282, 423, -420, 630, -1660, 2490, -1304, 1956, -11332, 16998, 3896, -5844, -95240, 142860, 157160, -235740, -983610, 1475415, 2634300, -3951450, -11751660, 17627490, 38381160, -57571740

Offset: 0

Author

Michael Somos, Jul 27 2003

Keywords

• sign

Comments

Series reversion of Sum_{k>=0} a(k)x^k is x(Sum_{k>=0} A007051(k)x^k).

G.f. A(x) = Sum_{k>=0} a(k)x^k satisfies 0 = x - (4x+1)*A(x) + (3x+2)*A(x)^2.

If A(x)=g.f., then y=x/A(x)-2x satisfies x^2 = y^2 - y.

Examples

			a(5) = 6 = upper left term of (-1)*M^5. - _Gary W. Adamson_, Jan 05 2012
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Programs

Mathematica

CoefficientList[Series[(1 + 4 x - Sqrt[1 + 4 x^2])/(4 + 6 x), {x, 0, 50}], x] (* Harvey P. Dale, Mar 24 2011 *)

PARI

a(n)=polcoeff((1+4*x-sqrt(1+4*x^2+x*O(x^n)))/(4+6*x),n)

Formula

G.f.: (1+4x-sqrt(1+4x^2))/(4+6x).

G.f.: (x-x^2*c(-x^2))/(1+x-x^2*c(-x^2)), c(x) the g.f. of A000108. - Paul Barry, Jun 17 2005

From Gary W. Adamson, Jan 05 2012: (Start)

a(n) is the upper left term of (-1)*M^n, where M = an infinite square production matrix as follows:

-1, -1, 0, 0, 0, 0, ...

-1, 1, -1, 0, 0, 0, ...

-1, 1, 1, -1, 0, 0, ...

-1, 1, 1, 1, -1, 0, ...

-1, 1, 1, 1, 1, -1, ...

... (End)

D-finite with recurrence 2*n*a(n) +3*n*a(n-1) +8*(n-3)*a(n-2) +12*(n-3)*a(n-3)=0. - R. J. Mathar, Nov 24 2012

Lim sup n->infinity |a(n)|^(1/n) = 2. - Vaclav Kotesovec, Feb 09 2014