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A032031 Triple factorial numbers: (3n)!!! = 3^n*n!. 41
1, 3, 18, 162, 1944, 29160, 524880, 11022480, 264539520, 7142567040, 214277011200, 7071141369600, 254561089305600, 9927882482918400, 416971064282572800, 18763697892715776000, 900657498850357248000, 45933532441368219648000, 2480410751833883860992000 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

REFERENCES

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.

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..100

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

FORMULA

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

a(n) = prod(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

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 *)

PROG

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

(PARI) a(n)=3^n*n!; /* or: */ a(n)=prod(k=1, n, 3*k );

(Sage)

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

(Haskell)

a032031 n = a032031_list !! n

a032031_list = scanl (*) 1 $ tail a008585_list

-- Reinhard Zumkeller, Sep 20 2013

CROSSREFS

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

Cf. Subsequence of A007661.

Sequence in context: A052182 A115415 A065058 * A127646 A089466 A107403

Adjacent sequences:  A032028 A032029 A032030 * A032032 A032033 A032034

KEYWORD

nonn,easy,nice

AUTHOR

Christian G. Bower

STATUS

approved

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Last modified February 21 22:56 EST 2018. Contains 299427 sequences. (Running on oeis4.)