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A329965 a(n) = ((1+n)*floor(1+n/2))*(n!/floor(1+n/2)!)^2.

%I A329965 #13 Jun 19 2025 00:26:15
%S A329965 1,2,6,72,240,7200,25200,1411200,5080320,457228800,1676505600,
%T A329965 221298739200,821966745600,149597947699200,560992303872000,
%U A329965 134638152929280000,508633022177280000,155641704786247680000,591438478187741184000,224746621711341649920000
%N A329965 a(n) = ((1+n)*floor(1+n/2))*(n!/floor(1+n/2)!)^2.
%F A329965 a(n) = n!*A212303(n+1).
%F A329965 a(n) = (n+1)!*A057977(n).
%F A329965 a(n) = A093005(n+1)*A262033(n)^2.
%F A329965 a(n) = A093005(n+1)*A329964(n).
%F A329965 a(2*n) = A052510(n) (n >= 0).
%F A329965 a(2*n+1) = A123072(n+1) (n >= 0).
%F A329965 a(n) = n! [x^n] (1 - sqrt(1 - 4*x^2) - 4*x^2*(1 - x - sqrt(1 - 4*x^2)))/(2*x^2*(1 - 4*x^2)^(3/2)).
%p A329965 A329965 := n -> ((1+n)*floor(1+n/2))*(n!/floor(1+n/2)!)^2:
%p A329965 seq(A329965(n), n=0..19);
%t A329965 ser := Series[(1 - Sqrt[1 - 4 x^2] - 4 x^2 (1 - x - Sqrt[1 - 4 x^2]))/(2 x^2 (1 - 4 x^2)^(3/2)), {x, 0, 22}]; Table[n! Coefficient[ser, x, n], {n, 0, 20}]
%t A329965 Table[(1+n)Floor[1+n/2](n!/Floor[1+n/2]!)^2,{n,0,30}] (* _Harvey P. Dale_, Oct 01 2023 *)
%o A329965 (Python)
%o A329965 from fractions import Fraction
%o A329965 def A329965():
%o A329965     x, n = 1, Fraction(1)
%o A329965     while True:
%o A329965         yield int(x)
%o A329965         m = n if n % 2 else 4/(n+2)
%o A329965         n += 1
%o A329965         x *= m * n
%o A329965 a = A329965(); [next(a) for i in range(36)]
%Y A329965 Cf. A212303, A057977, A052510, A123072, A093005, A262033, A329964.
%K A329965 nonn
%O A329965 0,2
%A A329965 _Peter Luschny_, Dec 04 2019