A104551 Expansion of x/((1-x)*sqrt(1+4*x^2)).

0, 1, 1, -1, -1, 5, 5, -15, -15, 55, 55, -197, -197, 727, 727, -2705, -2705, 10165, 10165, -38455, -38455, 146301, 146301, -559131, -559131, 2145025, 2145025, -8255575, -8255575, 31861025, 31861025, -123256495, -123256495, 477823895, 477823895, -1855782325, -1855782325, 7219352975

Offset: 0

Author

Paul Barry, Mar 14 2005

Keywords

• easy
• sign

Comments

A transformation of the Fibonacci numbers A000045 by the Riordan array (1/sqrt(1+4*x^2), (sqrt(1+4*x^2)-1)/(2*x)).

Links

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

Crossrefs

Cf. A000045, A054108.

Programs

Magma

R:=PowerSeriesRing(Integers(), 40); Coefficients(R!( x/((1-x)*Sqrt(1+4*x^2)) )); // G. C. Greubel, Jan 01 2023

Mathematica

CoefficientList[Series[x/((1-x)*Sqrt[1+4*x^2]), {x,0,40}], x] (* G. C. Greubel, Jan 01 2023 *)

SageMath

def A104551_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x/((1-x)*sqrt(1+4*x^2)) ).list()
A104551_list(40) # G. C. Greubel, Jan 01 2023

Formula

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

a(n) = Sum_{k=0..n} (sin(Pi*k/2)+cos(Pi*k)/2+1/2)*C(k-1,(k-1)/2)*(1-(-1)^k)/2.

D-finite with recurrence: (n-1)*a(n) = (n-1)*a(n-1) - 4*(n-2)*a(n-2) + 4*(n-2)*a(n-3). - R. J. Mathar, Feb 20 2015