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 A001901 Successive numerators of Wallis's approximation to Pi/2 (reduced). 5
 1, 2, 4, 16, 64, 128, 256, 2048, 16384, 32768, 65536, 262144, 1048576, 2097152, 4194304, 67108864, 1073741824, 2147483648, 4294967296, 17179869184, 68719476736, 137438953472, 274877906944, 2199023255552 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 REFERENCES H.-D. Ebbinghaus et al., Numbers, Springer, 1990, p. 146. LINKS J. Sondow, A faster product for Pi and a new integral for ln(Pi/2), arXiv:math/0401406 [math.NT], 2004. J. Sondow, A faster product for Pi and a new integral for ln(Pi/2), Amer. Math. Monthly 112 (2005), 729-734 and 113 (2006), 670. Eric Weisstein's World of Mathematics, Pi Eric Weisstein's World of Mathematics, Pi Continued Fraction FORMULA (2*2*4*4*6*6*8*8*...*2n*2n*...)/(1*3*3*5*5*7*7*9*...*(2n-1)*(2n+1)*...) for n >= 1. From Wolfdieter Lang, Dec 07 2017: (Start) 1/1 * 2/1 * 2/3 * 4/3 * 4/5 * 6/5 * 6/7 * ...; partial products (reduced). Here the numerators with offset 0. a(n) = numerator(W(n)), for n >= 0, with W(n) = Product_{k=0..n} N(k)/D(k) (reduced), with N(k) = 2*floor((k+1)/2) for k >= 1 and N(0) = 1, and D(k) = 2*floor(k/2) + 1, for k >= 0. (End) EXAMPLE From Wolfdieter Lang, Dec 07 2017: (Start) The Wallis numerators (N) and denominators (D) with partial products A(n) = A001900(n) and B(n) = A000246(n+1) in unreduced form, and a(n) and b(n) = A001902(n) in reduced form. n, k:     0  1  2  3  4   5    6     7      8        9       10 ... N(k):     1  2  2  4  4   6    6     8      8       10       10 ... D(k):     1  1  3  3  5   5    7     7      9        9        9 ... A(n):     1  2  4 16 64 384 2304 18432 147456  1474560 14745600 ... B(n):     1  1  3  9 45 225 1575 11025  99225   893025  9823275 ... a(n):     1  2  4 16 64 128  256  2048  16384    32768    65536 ... b(n):     1  1  3  9 45  75  175  1225  11025    19845    43659 ... n = 5: numerator(1*2*2*4*4*6/(1*1*3*3*5*5)) = numerator(384/225) = numerator(128/75) = 128. (End) MATHEMATICA a[n_?EvenQ] := n!!^2/((n - 1)!!^2*(n + 1)); a[n_?OddQ] := ((n - 1)!!^2*(n + 1))/n!!^2; Table[a[n] // Numerator, {n, 0, 23}] (* Jean-François Alcover, Jun 19 2013 *) CROSSREFS Denominators are A001902. Subsequence of A000079. Sequence in context: A138867 A138868 A138871 * A127588 A187822 A092585 Adjacent sequences:  A001898 A001899 A001900 * A001902 A001903 A001904 KEYWORD nonn,frac,easy AUTHOR STATUS approved

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Last modified April 24 18:03 EDT 2019. Contains 322430 sequences. (Running on oeis4.)