A323219 a(n) = [x^n] (1 - 4*x)^(-n/2)*x/(1 - x).

0, 1, 5, 37, 313, 2811, 26093, 247311, 2377905, 23104441, 226289605, 2230309533, 22093913449, 219786279909, 2194096906461, 21969023675097, 220538907003489, 2218881134793411, 22368588800763701, 225891901214751423, 2284746661102951833, 23140953249273852519

Offset: 0

Author

Peter Luschny, Jan 26 2019

Keywords

• nonn

Crossrefs

Central diagonal of A323222.

Cf. A242798.

Programs

Maple

ogf := n -> (1 - 4*x)^(-n/2)*x/(1 - x):
ser := n -> series(ogf(n), x, 46):
seq(coeff(ser(n), x, n), n=0..21);

Formula

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

From Vaclav Kotesovec, Jan 29 2019: (Start)

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).

a(n) ~ 2^(n - 1/2) * 3^((3*n - 1)/2) / (5*sqrt(Pi*n)). (End)

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