A047053 a(n) = 4^n * n!.

1, 4, 32, 384, 6144, 122880, 2949120, 82575360, 2642411520, 95126814720, 3805072588800, 167423193907200, 8036313307545600, 417888291992371200, 23401744351572787200, 1404104661094367232000, 89862698310039502848000

Offset

0,2

Comments

Original name was "Quadruple factorial numbers".

For n >= 1, a(n) is the order of the wreath product of the symmetric group S_n and the Abelian group (C_4)^n. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 07 2001

Number of n X n monomial matrices with entries 0, +/-1, +/-i.

a(n) is the product of the positive integers <= 4*n that are multiples of 4. - Peter Luschny, Jun 23 2011

Also, a(n) is the number of signed permutations of length 2*n that are equal to their reverse-complements. (See the Hardt and Troyka reference.) - Justin M. Troyka, Aug 13 2011.

Pi^n/a(n) is the volume of a 2*n-dimensional ball with radius 1/2. - Peter Luschny, Jul 24 2012

Equals the first right hand column of A167557, and also equals the first right hand column of A167569. - Johannes W. Meijer, Nov 12 2009

a(n) is the order of the group U_n(Z[i]) = {A in M_n(Z[i]): A*A^H = I_n}, the group of n X n unitary matrices over the Gaussian integers. Here A^H is the conjugate transpose of A. - Jianing Song, Mar 29 2021

Links

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

R. B. Brent, Generalizing Tuenter's Binomial Sums, J. Int. Seq. 18 (2015), # 15.3.2.

CombOS - Combinatorial Object Server, Generate colored permutations

R. Coquereaux and J.-B. Zuber, Maps, immersions and permutations, arXiv preprint arXiv:1507.03163 [math.CO], 2015-2016. Also J. Knot Theory Ramifications 25, 1650047 (2016), DOI.

Sylvie Corteel and Lauren Williams, Tableaux Combinatorics for the Asymmetric Exclusion Process II, arXiv:0810.2916 [math.CO], 2008-2009. [From Jonathan Vos Post, Oct 17 2008]

A. Hardt and J. M. Troyka, Restricted symmetric signed permutations, Pure Mathematics and Applications, Vol. 23 (No. 3, 2012), pp. 179-217.

A. Hardt and J. M. Troyka, Slides (associated with the Hardt and Troyka reference above).

INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 492.

Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, 5 (2002), Article 02.1.7.

M. D. Schmidt, Generalized j-Factorial Functions, Polynomials, and Applications, J. Int. Seq. 13 (2010), Article 10.6.7, p. 39.

Michael Z. Spivey and Laura L. Steil, The k-Binomial Transforms and the Hankel Transform, Journal of Integer Sequences, 9 (2006), Article 06.1.1.

Formula

a(n) = 4^n * n!.

E.g.f.: 1/(1 - 4*x).

Integral representation as the n-th moment of a positive function on a positive half-axis: in Maple notation a(n) = int(x^n*exp(-4*x)/4, x=0..infinity), n = 0, 1, ... This representation is unique. - Karol A. Penson, Jan 28 2002

Sum_{k>=0} (-1)^k/(2*k + 1)^n = (-1)^n * n * (PolyGamma[n-1, 1/4] - PolyGamma[n-1, 3/4]) / a(n) for n > 0. - Joseph Biberstine (jrbibers(AT)indiana.edu), Jul 27 2006

a(n) = Sum_{k=0..n} C(n,k)*(2k)!*(2(n-k))!/(k!(n-k)!) = Sum_{k=0..n} C(n,k)*A001813(k)*A001813(n-k). - Paul Barry, May 04 2007

E.g.f.: With interpolated zeros, 1 + sqrt(pi)*x*exp(x^2)*erf(x). - Paul Barry, Apr 10 2010

From Gary W. Adamson, Jul 19 2011: (Start)

a(n) = sum of top row terms of M^n, M = an infinite square production matrix as follows:

2, 2, 0, 0, 0, 0, ...

4, 4, 4, 0, 0, 0, ...

6, 6, 6, 6, 0, 0, ...

8, 8, 8, 8, 8, 0, ...

... (End)

G.f.: 1/(1 - 4*x/(1 - 4*x/(1 - 8*x/(1 - 8*x/(1 - 12*x/(1 - 12*x/(1 - 16*x/1 - ... (continued fraction). - Philippe Deléham, Jan 08 2012

G.f.: 2/G(0), where G(k) = 1 + 1/(1 - 8*x*(k + 1)/(8*x*(k + 1) - 1 + 8*x*(k + 1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013

G.f.: 1/Q(0), where Q(k) = 1 - 4*x*(2*k + 1) - 16*x^2*(k + 1)^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Sep 28 2013

a(n) = A000142(n) * A000302(n). - Michel Marcus, Nov 28 2013

a(n) = A087299(2*n). - Michael Somos, Jan 03 2015

D-finite with recurrence: a(n) - 4*n*a(n-1) = 0. - R. J. Mathar, Jan 27 2020

From Amiram Eldar, Jun 25 2020: (Start)

Sum_{n>=0) 1/a(n) = e^(1/4) (A092042).

Sum_{n>=0) (-1)^n/a(n) = e^(-1/4) (A092616). (End)

Example

G.f. = 1 + 4*x + 32*x^2 + 384*x^3 + 6144*x^4 + 122880*x^5 + 2949120*x^6 + ...

Maple

A047053:= n -> mul(k, k = select(k-> k mod 4 = 0, [$1..4*n])): seq(A047053(n), n = 0.. 16); # Peter Luschny, Jun 23 2011

Mathematica

a[n_]:= With[{m=2n}, If[ m<0, 0, m!*SeriesCoefficient[1 +Sqrt[Pi]*x*Exp[x^2]*Erf[x], {x, 0, m}]]]; (* Michael Somos, Jan 03 2015 *)
Table[4^n n!, {n, 0, 20}] (* Harvey P. Dale, Sep 19 2021 *)

Prog

(PARI)  a(n)=4^n*n!;
(Magma) [4^n*Factorial(n): n in [0..20]]; // Vincenzo Librandi, Jul 20 2011

Crossrefs

Cf. A000142, A000165, A007696, A008545, A032031, A047058, A087299, A092042, A092616.

a(n)= A051142(n+1, 0) (first column of triangle).

Sequence in context: A341502 A303049 A369536 * A201594 A222412 A349601

Adjacent sequences: A047050 A047051 A047052 * A047054 A047055 A047056

Keyword

nonn,easy

Author

Joe Keane (jgk(AT)jgk.org)

Extensions

Edited by Karol A. Penson, Jan 22 2002

Status

approved