A092615 -id:A092615 - cp's OEIS frontend

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

Christian G. Bower

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

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.

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

Cf. Subsequence of A007661.

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

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

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

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 );
```

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

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)

A092041 Decimal expansion of cube root of e.

Original entry on oeis.org

1, 3, 9, 5, 6, 1, 2, 4, 2, 5, 0, 8, 6, 0, 8, 9, 5, 2, 8, 6, 2, 8, 1, 2, 5, 3, 1, 9, 6, 0, 2, 5, 8, 6, 8, 3, 7, 5, 9, 7, 9, 0, 6, 5, 1, 5, 1, 9, 9, 4, 0, 6, 9, 8, 2, 6, 1, 7, 5, 1, 6, 7, 0, 6, 0, 3, 1, 7, 3, 9, 0, 1, 5, 6, 4, 5, 9, 5, 1, 8, 4, 6, 9, 6, 9, 7, 8, 8, 8, 1, 7, 2, 9, 5, 8, 3, 0, 2, 2, 4

Offset: 1

Mohammad K. Azarian, Mar 27 2004

e^(1/3) maximizes the value of x^(c/(x^3)) for any real positive constant c, and minimizes for it for a negative constant, on the range x > 0. - A.H.M. Smeets, Aug 16 2018

			1.39561242508608952862812531960258683759790651519940...

G. C. Greubel, Table of n, a(n) for n = 1..10000
Index entries for transcendental numbers

Cf. A001113, A019774, A091933, A092615 (reciprocal).

evalf(root[3](exp(1))) ; # R. J. Mathar, Oct 03 2011

RealDigits[E^(1/3), 10, 100][[1]] (* Alonso del Arte, Sep 13 2011 *)

exp(1/3) \\ Charles R Greathouse IV, Sep 14 2011

Equals (729/1552)*(1 + Sum_{n>=1} (1 + n^5/3 + n/3)/(3^n*n!)). - Alexander R. Povolotsky, Sep 13 2011
Equals (1/2)*(1 + (4 + (7 + (10 + ...)/9)/6)/3) = 1 + (1 + (1 + (1 + ...)/9)/6)/3. - Rok Cestnik, Jan 19 2017
Equals lim_{x->0} (tan(x)/x)^(1/x^2). - Amiram Eldar, Jul 04 2022

A027617 Number of permutations of n elements containing a 3-cycle.

Original entry on oeis.org

0, 0, 0, 2, 8, 40, 200, 1400, 11200, 103040, 1030400, 11334400, 135766400, 1764963200, 24709484800, 370687116800, 5930993868800, 100826895769600, 1814871926067200, 34482566595276800, 689651331905536000, 14482682605174784000, 318619017313845248000

Offset: 0

Joe Keane (jgk(AT)jgk.org)

nonn

a(n)/n! is asymptotic to 1-e^(-1/3) = 1 - A092615. - Michel Marcus, Aug 08 2013

Alois P. Heinz, Table of n, a(n) for n = 0..200

Column k=3 of A293211.

nn=20;Range[0,nn]!CoefficientList[Series[1/(1-x)-Exp[-x^3/3]/(1-x), {x,0,nn}],x]  (* Geoffrey Critzer, Jan 23 2013 *)

a(n) = n! * (1 - sum(k=0, floor(n/3), (-1)^k/(k!*3^k) ) ); \\ Stéphane Rézel, Dec 11 2019

a(n) = n! * ( 1 - Sum_{k=0..floor(n/3)} (-1)^k / (3^k * k!) ).
E.g.f.: 1/(1-x) - exp(-x^3/3)/(1-x). - Geoffrey Critzer, Jan 23 2013
Recurrence: a(n) = n*a(n-1) - (n-2)*(n-1)*a(n-3) + (n-3)*(n-2)*(n-1)*a(n-4). - Vaclav Kotesovec, Aug 13 2013
Conjectures from Stéphane Rézel, Dec 11 2019: (Start)
Recurrence: a(n) = n*a(n-1), for n > 3 and n !== 0 (mod 3);
for k > 1, a(3*k) = a(3*k-1)*S(k)/S(k-1) where S(k) = 3*k*S(k-1) - (-1)^k with S(1) = 1.
(End)

More terms from Geoffrey Critzer, Jan 23 2013

A367730 Decimal expansion of BesselJ(0,2/sqrt(3)).

Original entry on oeis.org

6, 9, 3, 4, 3, 6, 7, 8, 8, 1, 7, 9, 1, 8, 3, 1, 9, 0, 0, 9, 7, 7, 6, 0, 4, 6, 3, 3, 3, 3, 5, 4, 3, 9, 3, 1, 9, 7, 3, 2, 0, 9, 9, 5, 6, 2, 5, 3, 8, 6, 6, 5, 5, 5, 0, 9, 3, 4, 4, 4, 6, 5, 8, 3, 6, 6, 9, 3, 2, 6, 0, 3, 5, 4, 9, 3, 3, 5, 5, 6, 4, 1, 2, 9, 9, 8, 2, 1, 2, 7, 3, 0, 3, 2, 9, 0, 1, 6, 3

Offset: 0

Ilya Gutkovskiy, Nov 28 2023

			0.69343678817918319009776046333354393...

Cf. A091681, A092615, A334380, A334383, A367729.

RealDigits[BesselJ[0, 2/Sqrt[3]], 10, 99][[1]]

besselj(0,2/sqrt(3)) \\ Michel Marcus, Nov 29 2023

Equals Sum_{k>=0} 1 / ((-3)^k * k!^2).

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

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A092041 Decimal expansion of cube root of e.

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A027617 Number of permutations of n elements containing a 3-cycle.

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A367730 Decimal expansion of BesselJ(0,2/sqrt(3)).

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