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

0, 1, 1, 4, 9, 64, 75, 256, 1225, 16384, 19845, 65536, 160083, 1048576, 1288287, 4194304, 41409225, 1073741824, 1329696225, 4294967296, 10667118605, 68719476736, 85530896451, 274877906944, 1371086188563, 17592186044416, 21972535073125, 70368744177664, 176021737014375

Offset: 0

Author

Peter Luschny, Feb 11 2025

Keywords

• nonn
• frac

Examples

			r(n) = 0, 1, 1/2, 4/3, 9/16, 64/45, 75/128, 256/175, 1225/2048, ...
		

Links

• Paolo Xausa, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A380950 (denominator), A380910, A380909, A019267 (asymptotic coefficients).

Cf. A161736, A056982, A124399, A161737, A069955, A038534, A278145, A001901.

Programs

Maple

r := n -> (n/2)*(Pi/2)^cos(Pi*(n-1))*((n/2-1/2)!/(n/2)!)^2:
a := n -> numer(simplify(r(n))): seq(a(n), n = 0..28);
# Alternative:
r := n -> ifelse(n <= 1, n, (n - 1)/(n*r(n - 1))):

Mathematica

Join[{0}, Numerator[FoldList[(#2 - 1)/(#2*#) &, Range[30]]]] (* Paolo Xausa, Feb 14 2025 *)

Formula

Product_{k=1..n} a(k) = A380910(n) / A380909(n).

r(n) = (n - 1)/(n*r(n - 1)) for n > 1.

numerator(r(2*n)) = A161736(n).

numerator(r(2*n+1)) = A056982(n).

numerator(r(2*n+1))/4^n = A124399(n).

denominator(r(2*n-2)) = A161737(n).

denominator(r(2*n+1)) = A069955(n).

denominator(r(2*n+1))/(2*n+1) = A038534(n).

denominator(r(2*n+2))/2 = A278145(n).

denominator(r(2*n+2))/2^(2*n+1) = A001901(n).

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) ...).