A120995 -id:A120995 - cp's OEIS frontend

A069955 Let W(n) = Product_{k=1..n} (1 - 1/4k^2), the partial Wallis product (lim_{n->oo} W(n) = 2/Pi); then a(n) = numerator(W(n)).

1, 3, 45, 175, 11025, 43659, 693693, 2760615, 703956825, 2807136475, 44801898141, 178837328943, 11425718238025, 45635265151875, 729232910488125, 2913690606794775, 2980705490751054825, 11912508103174630875, 190453061649520333125, 761284675790187924375

Offset: 0

Benoit Cloitre, Apr 27 2002

Equivalently, denominators in partial products of the following approximation to Pi: Pi = Product_{n>=1} 4*n^2/(4*n^2-1). Numerators are in A056982.

Orin J. Farrell and Bertram Ross, Solved Problems in Analysis, Dover, NY, 1971; p. 77.

Seiichi Manyama, Table of n, a(n) for n = 0..832
Boris Gourévitch, L'univers de Pi.

Not the same as A001902(n).
Cf. A056982 (denominators), A001790, A046161.
W(n)=(3/4)*(A120995(n)/A120994(n)), n>=1.

a[n_] := Numerator[Product[1 - 1/(4*k^2), {k, 1, n}]]; Array[a, 20, 0] (* Amiram Eldar, May 07 2025 *)

a(n) = numerator(prod(k=1, n, 1-1/(4*k^2))); \\ Michel Marcus, Oct 22 2016

a(n) = numerator(W(n)), where W(n) = (2*n)!*(2*n+1)!/((2^n)*n!)^4.
W(n) = (2*n+1)*(binomial(2*n,n)/2^(2*n))^2 = (2*n+1)*(A001790(n)/A046161(n))^2 in lowest terms.
a(n) = (-1)^n*A056982(n)*C(-1/2,n)*C(n+1/2,n). - Peter Luschny, Apr 08 2016

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

Original entry on oeis.org

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

Offset: 1

Wolfdieter Lang, Aug 01 2006

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).

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

W. Lang: Rationals r(n) and limit.

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).

cp's OEIS Frontend

A069955 Let W(n) = Product_{k=1..n} (1 - 1/4k^2), the partial Wallis product (lim_{n->oo} W(n) = 2/Pi); then a(n) = numerator(W(n)).

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