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A069955 Let W(n) = Prod_{k=1..n} 1-1/4k^2, the partial Wallis product ( lim n -> infinity W(n)=2/Pi ); then a(n) = numerator(W(n)). 3
1, 3, 45, 175, 11025, 43659, 693693, 2760615, 703956825, 2807136475, 44801898141, 178837328943, 11425718238025, 45635265151875, 729232910488125, 2913690606794775, 2980705490751054825, 11912508103174630875, 190453061649520333125, 761284675790187924375 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

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

LINKS

Seiichi Manyama, Table of n, a(n) for n = 0..832

B. Gourevitch, L'univers de Pi

FORMULA

a(n) = numerator(W(n)), with 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

PROG

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

CROSSREFS

Not the same as A001902(n).

Cf. A056982 (denominators).

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

Sequence in context: A071968 A093585 A062270 * A289193 A062346 A002682

Adjacent sequences:  A069952 A069953 A069954 * A069956 A069957 A069958

KEYWORD

nonn,frac,easy,changed

AUTHOR

Benoit Cloitre, Apr 27 2002

STATUS

approved

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Last modified December 16 09:12 EST 2017. Contains 296087 sequences.