|
|
A184120
|
|
Expansion of (1/(1+4x+2x^2))*c(x/(1+4x+2x^2)), c(x) the g.f. of A000108.
|
|
2
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,2
|
|
COMMENTS
|
Hankel transform is the (4,-3) Somos-4 sequence A184121.
|
|
LINKS
|
|
|
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
|
|
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 *)
|
|
PROG
|
(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
(Magma) m:=30; R<x>:=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
|
|
CROSSREFS
|
|
|
KEYWORD
|
sign,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|