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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)).

%I A069955 #36 May 08 2025 02:27:54
%S A069955 1,3,45,175,11025,43659,693693,2760615,703956825,2807136475,
%T A069955 44801898141,178837328943,11425718238025,45635265151875,
%U A069955 729232910488125,2913690606794775,2980705490751054825,11912508103174630875,190453061649520333125,761284675790187924375
%N 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)).
%C A069955 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.
%D A069955 Orin J. Farrell and Bertram Ross, Solved Problems in Analysis, Dover, NY, 1971; p. 77.
%H A069955 Seiichi Manyama, <a href="/A069955/b069955.txt">Table of n, a(n) for n = 0..832</a>
%H A069955 Boris Gourévitch, <a href="http://www.pi314.net">L'univers de Pi</a>.
%F A069955 a(n) = numerator(W(n)), where W(n) = (2*n)!*(2*n+1)!/((2^n)*n!)^4.
%F A069955 W(n) = (2*n+1)*(binomial(2*n,n)/2^(2*n))^2 = (2*n+1)*(A001790(n)/A046161(n))^2 in lowest terms.
%F A069955 a(n) = (-1)^n*A056982(n)*C(-1/2,n)*C(n+1/2,n). - _Peter Luschny_, Apr 08 2016
%t A069955 a[n_] := Numerator[Product[1 - 1/(4*k^2), {k, 1, n}]]; Array[a, 20, 0] (* _Amiram Eldar_, May 07 2025 *)
%o A069955 (PARI) a(n) = numerator(prod(k=1, n, 1-1/(4*k^2))); \\ _Michel Marcus_, Oct 22 2016
%Y A069955 Not the same as A001902(n).
%Y A069955 Cf. A056982 (denominators), A001790, A046161.
%Y A069955 W(n)=(3/4)*(A120995(n)/A120994(n)), n>=1.
%K A069955 nonn,frac,easy
%O A069955 0,2
%A A069955 _Benoit Cloitre_, Apr 27 2002