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A190903 a(n) = Product_{k in M_n} k, M_n = {k | 1 <= k <= 3n and k mod 3 = n mod 3}.

%I A190903 #19 May 14 2020 16:06:30
%S A190903 1,1,10,162,280,12320,524880,1106560,96342400,7142567040,17041024000,
%T A190903 2324549427200,254561089305600,664565853952000,126757680265216000,
%U A190903 18763697892715776000,52580450364682240000,13106744139423334400000,2480410751833883860992000
%N A190903 a(n) = Product_{k in M_n} k, M_n = {k | 1 <= k <= 3n and k mod 3 = n mod 3}.
%C A190903 For n > 0:
%C A190903 a(3*n)   = A032031(3*n) = 3^(3*n) * Gamma(3*n + 1).
%C A190903 a(3*n-1) = A008544(3*n-1) = 3^(3*n-1) * Gamma(3*n - 1/3) / Gamma(2/3).
%C A190903 a(3*n+1) = A007559(3*n+1) = 3^(3*n+3/2) * Gamma(3*n + 4/3) * Gamma(2/3) / (2*Pi).
%H A190903 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/Multifactorials">Multifactorials</a>
%F A190903 From _Johannes W. Meijer_, Jul 04 2011: (Start)
%F A190903 a(3*n+3)/(a(3*n)*a(3)) = A006566(n+1); Dodecahedral numbers
%F A190903 a(3*n+4)/a(3*n+1) = A136214(3*n+4, 3*n+1)
%F A190903 a(3*n+5)/a(3*n+2) = A112333(3*n+5, 3*n+2) (End)
%p A190903 A190903 := proc(n) local k; mul(k, k = select(k-> k mod 3 = n mod 3, [$1 .. 3*n])) end: seq(A190903(n), n=0..17);
%t A190903 a[n_] := Switch[Mod[n, 3], 0, 3^n Gamma[n+1], 2, 3^n Gamma[n+2/3]/ Gamma[2/3], 1, 3^(n-1) Gamma[n+1/3]/Gamma[4/3]] // Round;
%t A190903 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Jun 25 2019 *)
%o A190903 (PARI) a(n) = prod(k=1, 3*n, if (k % 3 == n % 3, k, 1)); \\ _Michel Marcus_, Jun 25 2019 and May 14 2020
%Y A190903 Cf. A190901.
%K A190903 nonn
%O A190903 0,3
%A A190903 _Peter Luschny_, Jul 03 2011