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A380949 a(n) = numerator(r(n)) where r(n) = (n/2)*(Pi/2)^cos(Pi*(n-1))*((n/2-1/2)!/(n/2)!)^2.

%I A380949 #21 Feb 14 2025 08:14:16
%S A380949 0,1,1,4,9,64,75,256,1225,16384,19845,65536,160083,1048576,1288287,
%T A380949 4194304,41409225,1073741824,1329696225,4294967296,10667118605,
%U A380949 68719476736,85530896451,274877906944,1371086188563,17592186044416,21972535073125,70368744177664,176021737014375
%N A380949 a(n) = numerator(r(n)) where r(n) = (n/2)*(Pi/2)^cos(Pi*(n-1))*((n/2-1/2)!/(n/2)!)^2.
%H A380949 Paolo Xausa, <a href="/A380949/b380949.txt">Table of n, a(n) for n = 0..1000</a>
%F A380949 Product_{k=1..n} a(k) = A380910(n) / A380909(n).
%F A380949 r(n) = (n - 1)/(n*r(n - 1)) for n > 1.
%F A380949 numerator(r(2*n)) = A161736(n).
%F A380949 numerator(r(2*n+1)) = A056982(n).
%F A380949 numerator(r(2*n+1))/4^n = A124399(n).
%F A380949 denominator(r(2*n-2)) = A161737(n).
%F A380949 denominator(r(2*n+1)) = A069955(n).
%F A380949 denominator(r(2*n+1))/(2*n+1) = A038534(n).
%F A380949 denominator(r(2*n+2))/2 = A278145(n).
%F A380949 denominator(r(2*n+2))/2^(2*n+1) = A001901(n).
%F A380949 r(n) ~ (2/Pi)^((-1)^n)*(1 - 1/(2*n) + 1/(8*n^2) + 1/(16*n^3) - 5/(128*n^4) - 23/(256*n^5) ...).
%e A380949 r(n) = 0, 1, 1/2, 4/3, 9/16, 64/45, 75/128, 256/175, 1225/2048, ...
%p A380949 r := n -> (n/2)*(Pi/2)^cos(Pi*(n-1))*((n/2-1/2)!/(n/2)!)^2:
%p A380949 a := n -> numer(simplify(r(n))): seq(a(n), n = 0..28);
%p A380949 # Alternative:
%p A380949 r := n -> ifelse(n <= 1, n, (n - 1)/(n*r(n - 1))):
%t A380949 Join[{0}, Numerator[FoldList[(#2 - 1)/(#2*#) &, Range[30]]]] (* _Paolo Xausa_, Feb 14 2025 *)
%Y A380949 Cf. A380950 (denominator), A380910, A380909, A019267 (asymptotic coefficients).
%Y A380949 Cf. A161736, A056982, A124399, A161737, A069955, A038534, A278145, A001901.
%K A380949 nonn,frac
%O A380949 0,4
%A A380949 _Peter Luschny_, Feb 11 2025