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A004732 Numerator of n!!/(n+3)!!.

%I A004732 #24 Dec 26 2021 20:59:33
%S A004732 1,1,2,1,8,5,16,7,128,21,256,33,1024,429,2048,715,32768,2431,65536,
%T A004732 4199,262144,29393,524288,52003,4194304,185725,8388608,334305,
%U A004732 33554432,9694845,67108864,17678835,2147483648
%N A004732 Numerator of n!!/(n+3)!!.
%D A004732 S. Janson, On the traveling fly problem, Graph Theory Notes of New York, Vol. XXXI, 17, 1996.
%H A004732 Robert Israel, <a href="/A004732/b004732.txt">Table of n, a(n) for n = 0..3327</a>
%H A004732 S. Janson, <a href="http://www2.math.uu.se/~svante/papers/sj114.pdf">On the traveling fly problem</a>
%F A004732 From _Robert Israel_, Jan 07 2019: (Start)
%F A004732 a(2*m) = 2^(2*m+1 - A048881(m) - A007814(m+1)).
%F A004732 a(2*m+1) = A000265(A000108(m+1)). (End)
%p A004732 f:= proc(n) local m,r;
%p A004732   m:= floor(n/2);
%p A004732   if n::even then 2^(2*m - padic:-ordp(binomial(2*m+1,m),2) - padic:-ordp(m+1,2))
%p A004732   else
%p A004732     r:= binomial(2*m+2,m+1)/(m+2);
%p A004732     r/2^padic:-ordp(r,2);
%p A004732 fi
%p A004732 end proc:
%p A004732 map(f, [$0..50]); # _Robert Israel_, Jan 07 2019
%t A004732 Numerator[Table[n!!/(n+3)!!,{n,0,40}]] (* _Harvey P. Dale_, Nov 26 2015 *)
%o A004732 (PARI) a(n) = numerator(prod(i=0, floor((n-1)/2), n-2*i)/prod(i=0, floor((n+2)/2), n+3-2*i)) \\ _Michel Marcus_, May 24 2013
%Y A004732 Cf. A000108, A000265, A004733, A007814, A048881.
%K A004732 nonn,frac
%O A004732 0,3
%A A004732 _N. J. A. Sloane_