A126079 - OEIS

%I #28 Sep 08 2022 08:45:29

%S 1,-4,2,0,-2,-8,-28,-96,-330,-1144,-4004,-14144,-50388,-180880,

%T -653752,-2377280,-8691930,-31935960,-117858900,-436698240,

%U -1623971580,-6059188080,-22676052360,-85100059200,-320188972740,-1207569840048,-4564276213608,-17286920538496,-65597689543400

%N G.f.: (1-2*x)*sqrt(1-4*x).

%H Vincenzo Librandi, <a href="/A126079/b126079.txt">Table of n, a(n) for n = 0..200</a>

%F From _Emeric Deutsch_, Mar 25 2007: (Start)

%F a(n) = binomial(2n,n) - 6*binomial(2n-2,n-1) + 8*binomial(2n-4,n-2).

%F a(n) = (3-n)*binomial(2n,n)/((2*n-3)*(2*n-1)). (End)

%F E.g.f.: 1 - 4*x + 2*x^2 - x^2*Q(0), where Q(k)= 1 - 2*x/(k+3 - (k+3)*(2*k+3)/(2*k+3 - (k+3)/Q(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, Apr 29 2013

%F a(n) ~ -2^(2*n-2)/(sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Jun 29 2013

%F D-finite with recurrence: n*a(n) +2*(-3*n+5)*a(n-1) +4*(2*n-7)*a(n-2)=0. - _R. J. Mathar_, Jan 23 2020

%p a:=n->(3-n)*binomial(2*n,n)/(2*n-3)/(2*n-1): seq(a(n),n=0..30); # _Emeric Deutsch_, Mar 25 2007

%p A126079List := proc(m) local A, P, n; A := [1,-4,2,0]; P := [-2,0];

%p for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-1]]);

%p A := [op(A), P[-1]] od; A end: A126079List(27); # _Peter Luschny_, Mar 26 2022

%t CoefficientList[Series[(1-2x)Sqrt[1-4x],{x,0,40}],x] (* _Harvey P. Dale_, Mar 28 2014 *)

%o (Magma) [(3-n)*Binomial(2*n,n)/((2*n-3)*(2*n-1)): n in [0..30]]; // _Vincenzo Librandi_, Mar 29 2014

%Y Cf. A126966, A126089.

%K sign

%O 0,2

%A _N. J. A. Sloane_, Mar 22 2007