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

1, 1, 10, 162, 280, 12320, 524880, 1106560, 96342400, 7142567040, 17041024000, 2324549427200, 254561089305600, 664565853952000, 126757680265216000, 18763697892715776000, 52580450364682240000, 13106744139423334400000, 2480410751833883860992000

Offset: 0

Author

Peter Luschny, Jul 03 2011

Keywords

• nonn

Comments

For n > 0:

a(3*n) = A032031(3*n) = 3^(3*n) * Gamma(3*n + 1).

a(3*n-1) = A008544(3*n-1) = 3^(3*n-1) * Gamma(3*n - 1/3) / Gamma(2/3).

a(3*n+1) = A007559(3*n+1) = 3^(3*n+3/2) * Gamma(3*n + 4/3) * Gamma(2/3) / (2*Pi).

Links

• Peter Luschny, Multifactorials

Crossrefs

Cf. A190901.

Programs

Maple

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);

Mathematica

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;
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Jun 25 2019 *)

PARI

a(n) = prod(k=1, 3*n, if (k % 3 == n % 3, k, 1)); \\ Michel Marcus, Jun 25 2019 and May 14 2020

Formula

From Johannes W. Meijer, Jul 04 2011: (Start)

a(3*n+3)/(a(3*n)*a(3)) = A006566(n+1); Dodecahedral numbers

a(3*n+4)/a(3*n+1) = A136214(3*n+4, 3*n+1)

a(3*n+5)/a(3*n+2) = A112333(3*n+5, 3*n+2) (End)