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A333306 a(n) = sqrt(Pi/4)*2^A048881(2*n)*L(2*n) where L(s) = lim_{t->s} (t/2)!/((1-t)/2)!.

%I A333306 #19 Apr 09 2021 20:35:16
%S A333306 1,1,-1,9,-45,1575,-42525,3274425,-42567525,5746615875,-488462349375,
%T A333306 102088631019375,-6431583754220625,1923043542511966875,
%U A333306 -336532619939594203125,136295711075535652265625,-3952575621190533915703125,2083007352367411373575546875
%N A333306 a(n) = sqrt(Pi/4)*2^A048881(2*n)*L(2*n) where L(s) = lim_{t->s} (t/2)!/((1-t)/2)!.
%F A333306 a(n) = Z(2*n)*A048896(2*n)/2 where Z(n) = Pi^n*(n*Zeta(1 - n))/((1 - n)*Zeta(n)) for n >= 1.
%F A333306 a(n) = (-1)^n*(2*n)!/((1 - 2*n)*A046161(2*n)).
%F A333306 A034386(2*n-2)/2 divides a(n), i.e., all odd primes <= 2*(n-1) divide a(n).
%F A333306 The number of distinct prime divisors of a(n) is A278617(n).
%p A333306 L := s -> limit((factorial(t/2)/factorial((1-t)/2)), t=s):
%p A333306 G := n -> 2^(add(i, i = convert(n+1, base, 2)) - 1): # A048896
%p A333306 a := s -> sqrt(Pi/4)*G(2*s)*L(2*s): seq(a(n), n=0..17);
%t A333306 A333306[n_] := (-1)^n ((2 n)!/(1 - 2 n)) 2^(-2 n + DigitCount[2 n, 2, 1]);
%t A333306 Table[A333306[n], {n, 0, 17}]
%Y A333306 Cf. A048881, A048896, A046161, A034386, A278617.
%K A333306 sign
%O A333306 0,4
%A A333306 _Peter Luschny_, May 17 2020