A120994 Numerators of rationals related to John Wallis' product formula for Pi/2 from his 'Arithmetica infinitorum' from 1659.

1, 16, 192, 4096, 16384, 262144, 1048576, 268435456, 3221225472, 17179869184, 68719476736, 13194139533312, 17592186044416, 281474976710656, 1125899906842624, 1152921504606846976, 4611686018427387904

Offset: 1

Author

Wolfdieter Lang, Aug 01 2006

Keywords

• nonn
• easy
• frac

Comments

The corresponding denominators are given in A120995.

The normalized sequence of rationals r(n):=(3/4)*W(n), with r(1)=1, converges to 3*Pi/8 = 1.178097245...

The product formula for Pi/2 of Wallis can be written like lim_{n to infinity} W(n) with the rationals W(n):=(((2*n)!!/(2*n-1)!!)^2)/(2*n+1) with the double factorials (2*n)!! = A000165(n) and (2*n-1)!! = A001147(n).

Examples

			Rationals r(n)=((3/4)*W(n)): [1, 16/15, 192/175, 4096/3675,
16384/14553, 262144/231231, 1048576/920205, 268435456/234652275,...]
		

Links

• W. Lang: Rationals r(n) and limit.

Formula

a(n) = numerator((3/4)*W(n)), n>=1, with the rationals W(n) given above. An equivalent form is W(n) = (((4^n)/binomial(2*n,n))^2)/(2*n+1).