A360185 - OEIS

%I #20 Mar 12 2023 11:15:47

%S 1,2,5,18,65,234,859,3198,12011,45422,172745,660010,2531411,9740590,

%T 37585189,145376930,563495201,2188229290,8511640099,33157034510,

%U 129334888721,505100839930,1974764074999,7728329887670,30272839608101,118682276550082,465645693340003

%N a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(2*n-4*k,n-2*k).

%H Seiichi Manyama, <a href="/A360185/b360185.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: 1 / ( sqrt(1-4*x) * (1 + x^2) ).

%F a(n) ~ 2^(2*n + 4) / (17*sqrt(Pi*n)). - _Vaclav Kotesovec_, Jan 29 2023

%F D-finite with recurrence n*a(n) +2*(-2*n+1)*a(n-1) +n*a(n-2) +2*(-2*n+1)*a(n-3)=0. - _R. J. Mathar_, Mar 12 2023

%F a(n)+a(n-2) = A000984(n). - _R. J. Mathar_, Mar 12 2023

%p A360185 := proc(n)

%p     add((-1)^k*binomial(2*n-4*k,n-2*k),k=0..n/2) ;

%p end proc:

%p seq(A360185(n),n=0..70) ; # _R. J. Mathar_, Mar 12 2023

%o (PARI) a(n) = sum(k=0, n\2, (-1)^k*binomial(2*n-4*k, n-2*k));

%o (PARI) my(N=30, x='x+O('x^N)); Vec(1/(sqrt(1-4*x)*(1+x^2)))

%Y Cf. A054108, A360186.

%K nonn,easy

%O 0,2

%A _Seiichi Manyama_, Jan 29 2023