A271318 - OEIS

%I #18 Jan 30 2020 21:29:17

%S 1,1,2,5,9,16,33,67,126,239,477,946,1809,3471,6858,13515,26001,50100,

%T 98577,193705,373734,721691,1416933,2779780,5372001,10386841,20366802,

%U 39915887,77216409,149422384,292749633,573363693,1109898126

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

%C a(365) is negative. - _Vaclav Kotesovec_, Apr 04 2016

%H Vaclav Kotesovec, <a href="/A271318/b271318.txt">Table of n, a(n) for n = 0..1000</a>

%F a(n) = Sum_{k=0..n} Sum_{j=floor(n/2)..(n+k)/2} 4^(j-k)*binomial(k,2*j-n)*binomial((2*j-n)/2,j-k).

%F D-finite with recurrence: (n-1)*a(n) = -(n-13)*a(n-2) + 3*(5*n-17)*a(n-4) + 12*(n-4)*a(n-6). - _Vaclav Kotesovec_, Apr 04 2016

%F a(n) ~ (1 + 1/sqrt(21))/2 * ((3 + sqrt(21))/2)^(n/2) if n is even and a(n) ~ (-1)^((n+1)/2) * 2^(n+7/2) / (25*sqrt(Pi)*n^(3/2)) if n is odd. - _Vaclav Kotesovec_, Apr 04 2016

%t CoefficientList[Series[1/(-x*Sqrt[4*x^2+1]-x^2+1), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Apr 04 2016 *)

%o (Maxima) a(n):=sum(sum(4^(j-k)*binomial(k,2*j-n)*binomial((2*j-n)/2,j-k),j,floor(n/2),(n+k)/2),k,0,n);

%o (PARI) x='x+O('x^99); Vec(1/(-x*sqrt(4*x^2+1)-x^2+1)) \\ _Altug Alkan_, Apr 04 2016

%K sign,easy

%O 0,3

%A _Vladimir Kruchinin_, Apr 04 2016