%I #15 Oct 27 2021 09:45:01
%S 0,1,5,37,313,2811,26093,247311,2377905,23104441,226289605,2230309533,
%T 22093913449,219786279909,2194096906461,21969023675097,
%U 220538907003489,2218881134793411,22368588800763701,225891901214751423,2284746661102951833,23140953249273852519
%N a(n) = [x^n] (1 - 4*x)^(-n/2)*x/(1 - x).
%F a(n) = 1/(-3)^(n/2) - 4^n * Pochhammer(n/2,n)/n! * hypergeom([1,3*n/2],[n+1],4). - _Robert Israel_, Jan 28 2019
%F From _Vaclav Kotesovec_, Jan 29 2019: (Start)
%F Recurrence: 3*(n-2)*(n-1)*(65*n - 213)*a(n) = (20995*n^3 - 152844*n^2 + 347783*n - 238614)*a(n-2) + 12*(3*n - 10)*(3*n - 8)*(65*n - 83)*a(n-4).
%F a(n) ~ 2^(n - 1/2) * 3^((3*n - 1)/2) / (5*sqrt(Pi*n)). (End)
%F G.f.: -(24*x*cos(arcsin(216*x^2-1)/3))/(sqrt(3-324*x^2)*(2*sin(arcsin(216*x^2-1)/3)-11)). - _Vladimir Kruchinin_, Oct 27 2021
%p ogf := n -> (1 - 4*x)^(-n/2)*x/(1 - x):
%p ser := n -> series(ogf(n), x, 46):
%p seq(coeff(ser(n), x, n), n=0..21);
%Y Central diagonal of A323222.
%Y Cf. A242798.
%K nonn
%O 0,3
%A _Peter Luschny_, Jan 26 2019