A099324 - OEIS

%I #22 Jul 23 2020 19:21:47

%S 1,0,-1,3,-8,22,-64,196,-625,2055,-6917,23713,-82499,290511,-1033411,

%T 3707851,-13402696,48760366,-178405156,656043856,-2423307046,

%U 8987427466,-33453694486,124936258126,-467995871776,1757900019100,-6619846420552,24987199492704,-94520750408708

%N Expansion of (1 + sqrt(1 + 4x))/(2(1 + x)).

%C Binomial transform is A099323. Second binomial transform is A072100.

%C Hankel transform is A049347. - _Paul Barry_, Aug 10 2009

%H Robert Israel, <a href="/A099324/b099324.txt">Table of n, a(n) for n = 0..1668</a>

%H Paul Barry, <a href="https://arxiv.org/abs/2004.04577">On a Central Transform of Integer Sequences</a>, arXiv:2004.04577 [math.CO], 2020.

%F a(n) = Sum_{k=0..2n} (2*0^(2n-k)-1)*C(k,floor(k/2)). - _Paul Barry_, Aug 10 2009

%F |a(n+2)| = A091491(n+2,2). - _Philippe Deléham_, Nov 25 2009

%F G.f.: T(0)/(2+2*x)), where T(k) = k+2 - 2*x*(2*k+1) + 2*x*(k+2)*(2*k+3)/T(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Nov 27 2013

%F D-finite with recurrence: (2+4*n)*a(n) + (4+5*n)*a(n+1) + (n+2)*a(n+2) = 0. - _Robert Israel_, Mar 27 2018

%p f:= gfun:-rectoproc({(2+4*n)*a(n)+(4+5*n)*a(n+1)+(n+2)*a(n+2), a(0) = 1, a(1) = 0}, a(n), remember):

%p map(f, [$0..50]); # _Robert Israel_, Mar 27 2018

%t CoefficientList[Series[(1+Sqrt[1+4x])/(2(1+x)),{x,0,40}],x] (* _Harvey P. Dale_, Jan 30 2014 *)

%Y Cf. A014138.

%K easy,sign

%O 0,4

%A _Paul Barry_, Oct 12 2004