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A004731
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Denominator of n!!/(n+1)!!. Also numerator of rational part of Haar measure on Grassmannian space G(n,1).
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7
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1, 1, 1, 2, 3, 8, 15, 16, 35, 128, 315, 256, 693, 1024, 3003, 2048, 6435, 32768, 109395, 65536, 230945, 262144, 969969, 524288, 2028117, 4194304, 16900975, 8388608, 35102025, 33554432, 145422675, 67108864
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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COMMENTS
| Also rational part of numerator of GAMMA(n/2+1)/GAMMA(n/2+1/2) (cf. A036039).
Contribution from Joseph E. Cooper III (easonrevant(AT)gmail.com), Nov 07 2010: (Start)
a(n) is equal to the numerator of x(n) defined by the recurrence:
x(n) = x(n-2) + 1/x(n-1), x(1) = 1, x(2) = 1. (End)
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REFERENCES
| S. Janson, On the traveling fly problem, Graph Theory Notes of New York, Vol. XXXI, 17, 1996.
D. A. Klain and G.-C. Rota, Introduction to Geometric Probability, Cambridge, p. 67.
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..302
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EXAMPLE
| 1, 1, (1/2)*Pi, 2, (3/4)*Pi, 8/3, (15/16)*Pi, 16/5, (35/32)*Pi, 128/35, (315/256)*Pi, ...
The sequence GAMMA(n/2+1)/GAMMA(n/2+1/2), n >= 0, begins 1/Pi^(1/2), 1/2*Pi^(1/2), 2/Pi^(1/2), 3/4*Pi^(1/2), 8/3/Pi^(1/2), 15/16*Pi^(1/2), 16/5/Pi^(1/2), ...
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MAPLE
| if n mod 2 = 0 then k := n/2; 2*k*Pi*binomial(2*k-1, k)/4^k else k := (n-1)/2; 4^k/binomial(2*k, k); fi;
f:=n->simplify(GAMMA(n/2+1)/GAMMA(n/2+1/2));
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PROG
| (Haskell)
import Data.Ratio ((%), numerator)
a004731 0 = 1
a004731 n = a004731_list !! n
a004731_list = map numerator ggs where
ggs = 0 : 1 : zipWith (+) ggs (map (1 /) $ tail ggs) :: [Rational]
-- Reinhard Zumkeller, Dec 08 2011
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CROSSREFS
| Cf. A001803, A004730, A036069.
Cf. A036039, A046161, A001790, A001803, A101926.
Sequence in context: A007919 A205101 A069752 * A135354 A122412 A174019
Adjacent sequences: A004728 A004729 A004730 * A004732 A004733 A004734
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KEYWORD
| nonn,easy,nice,frac
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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