A243816 Expansion of (x*sqrt(4*x^2+1)-x)/(x*sqrt(-(2*sqrt(4*x^2+1)-x-2)/x) + sqrt(4*x^2+1)-x-1).

-1, 2, 0, 2, 5, 10, 27, 86, 264, 806, 2559, 8332, 27343, 90498, 302801, 1022074, 3472577, 11868242, 40786623, 140851104, 488490057, 1700694884, 5941890068, 20826229564, 73208513161, 258031793698, 911704655945

Offset

0,2

Links

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

Formula

a(n) = (Sum_{i=0..n/2} (-1)^i*binomial(2*(n-1)-4*i, n-2*i)*binomial(n-1, i))/(n-1), n > 1, a(0)=-1, a(1)=2.

a(n) ~ 4*(15/4)^n / (sqrt(255*Pi) * n^(3/2)). - Vaclav Kotesovec, Jun 15 2014

Conjecture D-finite with recurrence: 2*n*(n-1)*(2*n-3)*a(n) -(n-1)*(47*n^2-221*n+260)*a(n-1) +4*(38*n^3-300*n^2+777*n-660)*a(n-2) +4*(-94*n^3+1006*n^2-3617*n+4355)*a(n-3) +16*(n-5)*(64*n^2-510*n+1041)*a(n-4) -16*(n-5)*(n-6)*(47*n-221)*a(n-5) +1920*(n-5)*(n-6)*(n-7)*a(n-6)=0. - R. J. Mathar, Jan 25 2020

Mathematica

CoefficientList[Series[x*(-1 + Sqrt[1 + 4*x^2])/ (-1 + Sqrt[1 + 4*x^2] + x*(-1 + Sqrt[(2 + x - 2*Sqrt[1 + 4*x^2])/x])), {x, 0, 20}], x] (* Vaclav Kotesovec, Jun 15 2014 *)

Prog

(Maxima)
a(n):=if n=0 then -1 else if n=1 then 2 else sum((-1)^i*binomial(2*(n-1)-4*i, n-2*i)*binomial(n-1, i), i, 0, n/2)/(n-1);
(PARI) x='x+O('x^30); Vec((x*sqrt(4*x^2+1)-x)/(x*sqrt(-(2*sqrt(4*x^2+1)-x-2)/x)+sqrt(4*x^2+1)-x-1)) \\ G. C. Greubel, Oct 06 2018
(Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((x*Sqrt(4*x^2+1)-x)/(x*Sqrt(-(2*Sqrt(4*x^2+1)-x-2)/x)+Sqrt(4*x^2+1)-x-1))); // G. C. Greubel, Oct 06 2018

Crossrefs

Sequence in context: A059432 A256488 A175631 * A243159 A339327 A258144

Adjacent sequences: A243813 A243814 A243815 * A243817 A243818 A243819

Keyword

sign

Author

Vladimir Kruchinin, Jun 11 2014

Status

approved