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A001902
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Successive denominators of Wallis's approximation to Pi/2 (reduced).
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8
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1, 1, 3, 9, 45, 75, 175, 1225, 11025, 19845, 43659, 160083, 693693, 1288287, 2760615, 41409225, 703956825, 1329696225, 2807136475, 10667118605, 44801898141, 85530896451, 178837328943, 1371086188563, 11425718238025
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OFFSET
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0,3
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REFERENCES
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H.-D. Ebbinghaus et al., Numbers, Springer, 1990, p. 146.
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LINKS
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FORMULA
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(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.
1/1 * 2/1 * 2/3 * 4/3 * 4/5 * 6/5 * 6/7 * ...; partial products (reduced). Here the denominators with offset 0.
a(n) = denominator(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)
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EXAMPLE
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See the table in A001901 for details.
n = 5: numerator(1*2*2*4*4*6/(1*1*3*3*5*5)) = denominator(384/225) = denominator(128/75) = 75. (End)
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MATHEMATICA
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a[n_?EvenQ] := n!!^2/((n - 1)!!^2*(n + 1)); a[n_?OddQ] := (n - 1)!!^2*(n + 1)/n!!^2; Table[a[n] // Denominator, {n, 0, 23}] (* Jean-François Alcover, Jun 19 2013 *)
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CROSSREFS
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KEYWORD
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nonn,frac,easy
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AUTHOR
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STATUS
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approved
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