login
a(n) = (2+[n/2])*n!/((1+[n/2])*[n/2]!^2).
1

%I #20 Aug 20 2022 08:50:56

%S 2,2,3,9,8,40,25,175,84,756,294,3234,1056,13728,3861,57915,14300,

%T 243100,53482,1016158,201552,4232592,764218,17577014,2912168,72804200,

%U 11143500,300874500,42791040,1240940160,164812365,5109183315,636438060,21002455980,2463251010

%N a(n) = (2+[n/2])*n!/((1+[n/2])*[n/2]!^2).

%F a(n) = A056040(n)*(2+[n/2])/(1+[n/2]).

%F a(n) = A057977(n)*A008619(n+2).

%F a(2*n+1) = (n+2)*binomial(2*n+1, n+1) = A189911(2*n+1).

%F a(2*n-3) = n*binomial(2*n-3, n-1) = A097070(n) for n>=2.

%F a(2*n+2) = (n+3)*binomial(2*n+2, n+1)/(n+2) = A038665(n).

%F Sum_{n>=0} 1/a(n) = 16/3 - 40*Pi/(9*sqrt(3)) + 4*Pi^2/9. - _Amiram Eldar_, Aug 20 2022

%p a := n -> (2+iquo(n,2))*n!/((1+iquo(n,2))*iquo(n, 2)!^2):

%p seq(a(n), n=0..34);

%o (Sage)

%o def A275329():

%o x, n, k = 1, 1, 2

%o while True:

%o yield x * k

%o if is_odd(n):

%o x *= n

%o else:

%o k += 1

%o x = (x<<2)//(n+2)

%o n += 1

%o a = A275329(); print([next(a) for _ in range(37)])

%Y Cf. A038665, A056040, A057977, A097070, A189911.

%K nonn

%O 0,1

%A _Peter Luschny_, Sep 10 2016