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

Author

Benoit Cloitre, Apr 27 2002

Keywords

• nonn
• frac
• easy

Comments

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.

References

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

Links

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

Crossrefs

Not the same as A001902(n).

Cf. A056982 (denominators), A001790, A046161.

W(n)=(3/4)*(A120995(n)/A120994(n)), n>=1.

Programs

Mathematica

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

PARI

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

Formula

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