A001902 Successive denominators of Wallis's approximation to Pi/2 (reduced).

1, 1, 3, 9, 45, 75, 175, 1225, 11025, 19845, 43659, 160083, 693693, 1288287, 2760615, 41409225, 703956825, 1329696225, 2807136475, 10667118605, 44801898141, 85530896451, 178837328943, 1371086188563, 11425718238025

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• frac
• easy

Examples

			From _Wolfdieter Lang_, Dec 07 2017: (Start)
See the table in A001901 for details.
n = 5: numerator(1*2*2*4*4*6/(1*1*3*3*5*5)) = denominator(384/225) = denominator(128/75) = 75. (End)
		

References

• H.-D. Ebbinghaus et al., Numbers, Springer, 1990, p. 146.

Links

• J. Sondow, A faster product for Pi and a new integral for ln(Pi/2), arXiv:math/0401406 [math.NT], 2004.
• J. Sondow, A faster product for Pi and a new integral for ln(Pi/2), Amer. Math. Monthly 112 (2005), 729-734 and 113 (2006), 670.

Crossrefs

Numerators are A001901. For the unreduced form see A001900(n)/A000246(n+1), n >= 0.

Programs

Mathematica

a[n_?EvenQ] := n!!^2/((n - 1)!!^2*(n + 1)); a[n_?OddQ] := (n - 1)!!^2*(n + 1)/n!!^2; Table[a[n] // Denominator, {n, 0, 23}] (* Jean-François Alcover, Jun 19 2013 *)

Formula

(2*2*4*4*6*6*8*8*...*2n*2n*...)/(1*3*3*5*5*7*7*9*...*(2n-1)*(2n+1)*...) for n >= 1.

From Wolfdieter Lang, Dec 07 2017: (Start)

1/1 * 2/1 * 2/3 * 4/3 * 4/5 * 6/5 * 6/7 * ...; partial products (reduced). Here the denominators with offset 0.

a(n) = denominator(W(n)), for n >= 0, with W(n) = Product_{k=0..n} N(k)/D(k) (reduced), with N(k) = 2*floor((k+1)/2) for k >= 1 and N(0) = 1, and D(k) = 2*floor(k/2) + 1, for k >= 0. (End)

a(n) is the denominator of the continued fraction [1;1,1/2,1/3,...,1/n]. - Thomas Ordowski, Oct 19 2024