A120994 -id:A120994 - 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

A120995 Denominators of rationals related to John Wallis' product formula for Pi/2 (from his 'Arithmetica infinitorum' from 1659).

Original entry on oeis.org

1, 15, 175, 3675, 14553, 231231, 920205, 234652275, 2807136475, 14933966047, 59612442981, 11425718238025, 15211755050625, 243077636829375, 971230202264925, 993568496917018275, 3970836034391543625

Offset: 1

Wolfdieter Lang, Aug 01 2006

The corresponding numerators are A120994.

The rationals W(n):= (((4^n)/binomial(2*n,n))^2)/(2*n+1) converge to Pi/2 = 1.570796327...

a(n) = denominators((3/4)*W(n)), 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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