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Denominator of rational part of Haar measure on Grassmannian space G(n,1).
4

%I #16 Sep 23 2022 16:38:13

%S 1,2,1,4,3,16,5,32,35,256,63,512,231,2048,429,4096,6435,65536,12155,

%T 131072,46189,524288,88179,1048576,676039,8388608,1300075,16777216,

%U 5014575,67108864,9694845,134217728,300540195

%N Denominator of rational part of Haar measure on Grassmannian space G(n,1).

%C Also rational part of denominator of Gamma(n/2+1)/Gamma(n/2+1/2) (cf. A004731).

%D D. A. Klain and G.-C. Rota, Introduction to Geometric Probability, Cambridge, p. 67.

%H G. C. Greubel, <a href="/A036069/b036069.txt">Table of n, a(n) for n = 0..1000</a>

%e 1, 1, 1/2*Pi, 2, 3/4*Pi, 8/3, 15/16*Pi, 16/5, 35/32*Pi, 128/35, 315/256*Pi, ...

%e 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), ...

%p 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;

%p f:=n->simplify(GAMMA(n/2+1)/GAMMA(n/2+1/2));

%t Table[ Denominator[ Gamma[n/2+1]/Gamma[n/2+1/2]*Sqrt[Pi]^(1 - 2 Mod[n, 2])], {n, 0, 32}] (* _Jean-François Alcover_, Jul 16 2012 *)

%Y Cf. A004731.

%Y Bisections are A001790 and A101926.

%Y Cf. A004731, A046161, A001790, A001803, A101926.

%K nonn,easy,nice,frac

%O 0,2

%A _N. J. A. Sloane_