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

%I A032031 #81 Aug 07 2024 10:08:27
%S A032031 1,3,18,162,1944,29160,524880,11022480,264539520,7142567040,
%T A032031 214277011200,7071141369600,254561089305600,9927882482918400,
%U A032031 416971064282572800,18763697892715776000,900657498850357248000,45933532441368219648000,2480410751833883860992000
%N A032031 Triple factorial numbers: (3n)!!! = 3^n*n!.
%C A032031 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
%C A032031 Laguerre transform of double factorials 2^n*n! = A000165(n). - _Paul Barry_, Aug 08 2008
%C A032031 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
%C A032031 a(n) is the product of the positive integers <= 3*n that are multiples of 3. - _Peter Luschny_, Jun 23 2011
%C A032031 Partial products of A008585. - _Reinhard Zumkeller_, Sep 20 2013
%H A032031 Vincenzo Librandi, <a href="/A032031/b032031.txt">Table of n, a(n) for n = 0..100</a>
%H A032031 CombOS - Combinatorial Object Server, <a href="http://combos.org/cperm">Generate colored permutations</a>
%H A032031 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=491">Encyclopedia of Combinatorial Structures 491</a>
%H A032031 Alexsandar Petojevic, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">The Function vM_m(s; a; z) and Some Well-Known Sequences</a>, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
%H A032031 Michael Z. Spivey and Laura L. Steil, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL9/Spivey/spivey7.html">The k-Binomial Transforms and the Hankel Transform</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.1.1.
%F A032031 a(n) = 3^n*n!.
%F A032031 a(n) = Product_{k=1..n} 3*k.
%F A032031 E.g.f.: 1/(1-3*x).
%F A032031 a(n) = Sum_{k=0..n} C(n,k)*(n!/k!)*2^k*k!. - _Paul Barry_, Aug 08 2008
%F A032031 a(0) = 1, a(n) = 3*n*a(n-1). - _Arkadiusz Wesolowski_, Oct 04 2011
%F A032031 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
%F A032031 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
%F A032031 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
%F A032031 From _Amiram Eldar_, Jun 25 2020: (Start)
%F A032031 Sum_{n>=0} 1/a(n) = e^(1/3) (A092041).
%F A032031 Sum_{n>=0} (-1)^n/a(n) = e^(-1/3) (A092615). (End)
%p A032031 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
%p A032031 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
%t A032031 Table[3^n*Gamma[1 + n], {n, 0, 20}] (* _Roger L. Bagula_, Oct 30 2008 *)
%t A032031 Join[{1},FoldList[Times,3*Range[20]]] (* _Harvey P. Dale_, Feb 10 2019 *)
%t A032031 Table[Times@@Range[3n,1,-3],{n,0,20}] (* _Harvey P. Dale_, Apr 14 2023 *)
%o A032031 (Magma) [3^n*Factorial(n): n in [0..60]]; // _Vincenzo Librandi_, Apr 22 2011
%o A032031 (PARI) a(n)=3^n*n!;
%o A032031 (PARI) a(n)=prod(k=1,n, 3*k );
%o A032031 (SageMath)
%o A032031 def A032031(n) : return mul(j for j in range(3,3*(n+1),3))
%o A032031 [A032031(n) for n in (0..16)]  # _Peter Luschny_, May 20 2013
%o A032031 (Haskell)
%o A032031 a032031 n = a032031_list !! n
%o A032031 a032031_list = scanl (*) 1 $ tail a008585_list
%o A032031 -- _Reinhard Zumkeller_, Sep 20 2013
%Y A032031 Cf. A000142, A007559, A008544, A051141, A000165, A092041, A092615.
%Y A032031 Cf. Subsequence of A007661.
%K A032031 nonn,easy,nice
%O A032031 0,2
%A A032031 _Christian G. Bower_