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A275329 a(n) = (2+[n/2])*n!/((1+[n/2])*[n/2]!^2).

%I A275329 #20 Aug 20 2022 08:50:56
%S A275329 2,2,3,9,8,40,25,175,84,756,294,3234,1056,13728,3861,57915,14300,
%T A275329 243100,53482,1016158,201552,4232592,764218,17577014,2912168,72804200,
%U A275329 11143500,300874500,42791040,1240940160,164812365,5109183315,636438060,21002455980,2463251010
%N A275329 a(n) = (2+[n/2])*n!/((1+[n/2])*[n/2]!^2).
%F A275329 a(n) = A056040(n)*(2+[n/2])/(1+[n/2]).
%F A275329 a(n) = A057977(n)*A008619(n+2).
%F A275329 a(2*n+1) = (n+2)*binomial(2*n+1, n+1) = A189911(2*n+1).
%F A275329 a(2*n-3) = n*binomial(2*n-3, n-1) = A097070(n) for n>=2.
%F A275329 a(2*n+2) = (n+3)*binomial(2*n+2, n+1)/(n+2) = A038665(n).
%F A275329 Sum_{n>=0} 1/a(n) = 16/3 - 40*Pi/(9*sqrt(3)) + 4*Pi^2/9. - _Amiram Eldar_, Aug 20 2022
%p A275329 a := n -> (2+iquo(n,2))*n!/((1+iquo(n,2))*iquo(n, 2)!^2):
%p A275329 seq(a(n), n=0..34);
%o A275329 (Sage)
%o A275329 def A275329():
%o A275329     x, n, k = 1, 1, 2
%o A275329     while True:
%o A275329         yield x * k
%o A275329         if is_odd(n):
%o A275329             x *= n
%o A275329         else:
%o A275329             k += 1
%o A275329             x = (x<<2)//(n+2)
%o A275329         n += 1
%o A275329 a = A275329(); print([next(a) for _ in range(37)])
%Y A275329 Cf. A038665, A056040, A057977, A097070, A189911.
%K A275329 nonn
%O A275329 0,1
%A A275329 _Peter Luschny_, Sep 10 2016