A184120 Expansion of (1/(1+4x+2x^2))*c(x/(1+4x+2x^2)), c(x) the g.f. of A000108.

1, -3, 8, -23, 70, -218, 688, -2195, 7062, -22866, 74416, -243206, 797660, -2624004, 8654304, -28607171, 94748774, -314361682, 1044625200, -3476135186, 11581870900, -38632753228, 128998096032, -431144781486, 1442252806012

Offset: 0

Author

Paul Barry, Jan 09 2011

Keywords

• sign
• easy

Comments

Hankel transform is the (4,-3) Somos-4 sequence A184121.

Links

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

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((Sqrt(2*x^2+4*x+1)-Sqrt(2*x^2+1))/( 2*x*Sqrt(2*x^2+4*x+1)))); // G. C. Greubel, Aug 14 2018

Mathematica

CoefficientList[Series[(Sqrt[2x^2+4x+1]-Sqrt[2x^2+1])/(2x Sqrt[2x^2+4x+1]),{x,0,30}],x] (* Harvey P. Dale, Mar 09 2012 *)

PARI

x='x+O('x^30); Vec((sqrt(2*x^2+4*x+1)-sqrt(2*x^2+1))/( 2*x*sqrt(2*x^2+4*x+1))) \\ G. C. Greubel, Aug 14 2018

Formula

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

G.f.: 1/(1+4x+2x^2-x/(1-x/(1+4x+2x^2-x/(1-x/(1+4x+2x^2-x/(1-x/(1-... (continued fraction).

Conjecture: (n+1)*a(n) +2*(2n+1)*a(n-1) +4*(n-1)*a(n-2) +4*(2n-5)*a(n-3) +4*(n-3)*a(n-4)=0. - R. J. Mathar, Nov 17 2011

G.f.: 1/(2*x) - G(0)/(2*x), where G(k)= 1 - 4*x*(4*k+1)/( (1+2*x^2)*(4*k+2) - x*(1+2*x^2)*(4*k+2)*(4*k+3)/(x*(4*k+3) - (1+2*x^2)*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 26 2013

a(n) ~ (-1)^n * (2+sqrt(2))^n / (sqrt(3*sqrt(2)-4) * sqrt(Pi*n)). - Vaclav Kotesovec, Feb 04 2014