A032031 Triple factorial numbers: (3n)!!! = 3^n*n!.

1, 3, 18, 162, 1944, 29160, 524880, 11022480, 264539520, 7142567040, 214277011200, 7071141369600, 254561089305600, 9927882482918400, 416971064282572800, 18763697892715776000, 900657498850357248000, 45933532441368219648000, 2480410751833883860992000

Offset: 0

Author

Christian G. Bower

Keywords

• nonn
• easy
• nice

Comments

For n >= 1 a(n) is the order of the wreath product of the symmetric group S_n and the elementary Abelian group (C_3)^n. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 07 2001

Laguerre transform of double factorials 2^n*n! = A000165(n). - Paul Barry, Aug 08 2008

For positive n, a(n) equals the permanent of the n X n matrix consisting entirely of 3's. - John M. Campbell, May 26 2011

a(n) is the product of the positive integers <= 3*n that are multiples of 3. - Peter Luschny, Jun 23 2011

Partial products of A008585. - Reinhard Zumkeller, Sep 20 2013

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..100
• CombOS - Combinatorial Object Server, Generate colored permutations
• INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 491
• Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
• Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.

Crossrefs

Cf. A000142, A007559, A008544, A051141, A000165, A092041, A092615.

Cf. Subsequence of A007661.

Programs

Haskell

a032031 n = a032031_list !! n
a032031_list = scanl (*) 1 $ tail a008585_list
-- Reinhard Zumkeller, Sep 20 2013

Magma

[3^n*Factorial(n): n in [0..60]]; // Vincenzo Librandi, Apr 22 2011

Maple

with(combstruct):ZL:=[T,{T=Union(Z,Prod(Epsilon,Z,T), Prod(T,Z,Epsilon), Prod(T,Z))},labeled]:seq(count(ZL,size=i)/i,i=1..17); # Zerinvary Lajos, Dec 16 2007
A032031 := n -> mul(k, k = select(k-> k mod 3 = 0, [$1 .. 3*n])): seq(A032031(n), n = 0 .. 16); # Peter Luschny, Jun 23 2011

Mathematica

Table[3^n*Gamma[1 + n], {n, 0, 20}] (* Roger L. Bagula, Oct 30 2008 *)
Join[{1},FoldList[Times,3*Range[20]]] (* Harvey P. Dale, Feb 10 2019 *)
Table[Times@@Range[3n,1,-3],{n,0,20}] (* Harvey P. Dale, Apr 14 2023 *)

PARI

a(n)=3^n*n!;

PARI

a(n)=prod(k=1,n, 3*k );

SageMath

def A032031(n) : return mul(j for j in range(3,3*(n+1),3))
[A032031(n) for n in (0..16)]  # Peter Luschny, May 20 2013

Formula

a(n) = 3^n*n!.

a(n) = Product_{k=1..n} 3*k.

E.g.f.: 1/(1-3*x).

a(n) = Sum_{k=0..n} C(n,k)*(n!/k!)*2^k*k!. - Paul Barry, Aug 08 2008

a(0) = 1, a(n) = 3*n*a(n-1). - Arkadiusz Wesolowski, Oct 04 2011

G.f.: 2/G(0), where G(k)= 1 + 1/(1 - 6*x*(k+1)/(6*x*(k+1) - 1 + 6*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013

G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(3*k+3)/(x*(3*k+3) + 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 06 2013

G.f.: 1/Q(0), where Q(k) = 1 - 3*x*(2*k+1) - 9*x^2*(k+1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 28 2013

From Amiram Eldar, Jun 25 2020: (Start)

Sum_{n>=0} 1/a(n) = e^(1/3) (A092041).

Sum_{n>=0} (-1)^n/a(n) = e^(-1/3) (A092615). (End)