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a(n) = Product_{k in M_n} k; M_n = {k | 1 <= k <= 2n and k mod 2 = n mod 2}.
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%I #22 Feb 13 2017 03:34:50

%S 1,1,8,15,384,945,46080,135135,10321920,34459425,3715891200,

%T 13749310575,1961990553600,7905853580625,1428329123020800,

%U 6190283353629375,1371195958099968000,6332659870762850625,1678343852714360832000

%N a(n) = Product_{k in M_n} k; M_n = {k | 1 <= k <= 2n and k mod 2 = n mod 2}.

%H G. C. Greubel, <a href="/A190901/b190901.txt">Table of n, a(n) for n = 0..400</a>

%H Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/Multifactorials">Multifactorials</a>

%F a(2*k) = A006882(4*k) = 4^k * Gamma(2*k+1).

%F a(2*k+1) = A001147(2*k+1) = 4^k * Gamma(2*k+3/2) / sqrt(Pi/4).

%p A190901 := proc(n) local k; mul(k, k = select(k-> k mod 2 = n mod 2, [$1 .. 2*n])) end: seq(A190901(n), n=0..18);

%t a[n_] := With[{m = Mod[n, 2]}, Product[If[Mod[k, 2] == m, k, 1], {k, 1, 2*n}]]; Table[a[n], {n, 0, 18}] (* _Jean-François Alcover_, Jan 27 2014 *)

%K nonn

%O 0,3

%A _Peter Luschny_, Jun 23 2011