A360185 a(n) = Sum_{k=0..floor(n/2)} (-1)^k * binomial(2*n-4*k,n-2*k).

1, 2, 5, 18, 65, 234, 859, 3198, 12011, 45422, 172745, 660010, 2531411, 9740590, 37585189, 145376930, 563495201, 2188229290, 8511640099, 33157034510, 129334888721, 505100839930, 1974764074999, 7728329887670, 30272839608101, 118682276550082, 465645693340003

Offset: 0

Author

Seiichi Manyama, Jan 29 2023

Keywords

• nonn
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A054108, A360186.

Programs

Maple

A360185 := proc(n)
    add((-1)^k*binomial(2*n-4*k,n-2*k),k=0..n/2) ;
end proc:
seq(A360185(n),n=0..70) ; # R. J. Mathar, Mar 12 2023

PARI

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

PARI

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

Formula

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

a(n) ~ 2^(2*n + 4) / (17*sqrt(Pi*n)). - Vaclav Kotesovec, Jan 29 2023

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

a(n)+a(n-2) = A000984(n). - R. J. Mathar, Mar 12 2023