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%I A120994 #8 Aug 29 2019 17:36:39
%S A120994 1,16,192,4096,16384,262144,1048576,268435456,3221225472,17179869184,
%T A120994 68719476736,13194139533312,17592186044416,281474976710656,
%U A120994 1125899906842624,1152921504606846976,4611686018427387904
%N A120994 Numerators of rationals related to John Wallis' product formula for Pi/2 from his 'Arithmetica infinitorum' from 1659.
%C A120994 The corresponding denominators are given in A120995.
%C A120994 The normalized sequence of rationals r(n):=(3/4)*W(n), with r(1)=1, converges to 3*Pi/8 = 1.178097245...
%C A120994 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).
%H A120994 W. Lang: <a href="/A120994/a120994.txt">Rationals r(n) and limit.</a>
%F A120994 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).
%e A120994 Rationals r(n)=((3/4)*W(n)): [1, 16/15, 192/175, 4096/3675,
%e A120994 16384/14553, 262144/231231, 1048576/920205, 268435456/234652275,...]
%K A120994 nonn,easy,frac
%O A120994 1,2
%A A120994 _Wolfdieter Lang_, Aug 01 2006

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