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A036069 Denominator of rational part of Haar measure on Grassmannian space G(n,1).

%I A036069 #16 Sep 23 2022 16:38:13
%S A036069 1,2,1,4,3,16,5,32,35,256,63,512,231,2048,429,4096,6435,65536,12155,
%T A036069 131072,46189,524288,88179,1048576,676039,8388608,1300075,16777216,
%U A036069 5014575,67108864,9694845,134217728,300540195
%N A036069 Denominator of rational part of Haar measure on Grassmannian space G(n,1).
%C A036069 Also rational part of denominator of Gamma(n/2+1)/Gamma(n/2+1/2) (cf. A004731).
%D A036069 D. A. Klain and G.-C. Rota, Introduction to Geometric Probability, Cambridge, p. 67.
%H A036069 G. C. Greubel, <a href="/A036069/b036069.txt">Table of n, a(n) for n = 0..1000</a>
%e A036069 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 A036069 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 A036069 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 A036069 f:=n->simplify(GAMMA(n/2+1)/GAMMA(n/2+1/2));
%t A036069 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 A036069 Cf. A004731.
%Y A036069 Bisections are A001790 and A101926.
%Y A036069 Cf. A004731, A046161, A001790, A001803, A101926.
%K A036069 nonn,easy,nice,frac
%O A036069 0,2
%A A036069 _N. J. A. Sloane_