A190864 - OEIS

%I #24 Jan 30 2020 21:29:16

%S 1,1,1,3,5,5,9,21,29,31,65,143,181,183,441,1019,1165,893,2929,7829,

%T 7589,1677,19305,66585,49661,-44279,126881,638085,325525,-1024831,

%U 833049,6876389,2135149,-16612625,5467345,81608271,14007941,-244131809,35877321

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

%H Vincenzo Librandi, <a href="/A190864/b190864.txt">Table of n, a(n) for n = 0..100</a>

%F a(n)=sum(k=1..n, binomial(k/2,(n-k)/2)*2^(n-k-1)*((-1)^(n-k)+1)), n>0, a(0)=1.

%F D-finite with recurrence: (-n+1)*a(n) +(-n+2)*a(n-1) +3*(-n+5)*a(n-2) +3*(-n+6)*a(n-3) +4*(2*n-5)*a(n-4) +4*(2*n-7)*a(n-5) +16*(n-4)*a(n-6) +16*(n-5)*a(n-7)=0. - _R. J. Mathar_, Jan 25 2020

%t CoefficientList[Series[1/(1-x Sqrt[1+4x^2]),{x,0,40}],x] (* _Harvey P. Dale_, Mar 04 2015 *)

%o (PARI) x='x+O('x^66); /* that many terms */

%o Vec(1/(1-x*sqrt(1+4*x^2))) /* show terms */ /* Joerg Arndt, May 22 2011 */

%o (Maxima) a(n):=sum(binomial(k/2,(n-k)/2)*2^(n-k-1)*((-1)^(n-k)+1),k,1,n);

%K sign

%O 0,4

%A _Vladimir Kruchinin_, May 21 2011