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A047053 4^n*n!. 40
1, 4, 32, 384, 6144, 122880, 2949120, 82575360, 2642411520, 95126814720, 3805072588800, 167423193907200, 8036313307545600, 417888291992371200, 23401744351572787200, 1404104661094367232000, 89862698310039502848000 (list; graph; refs; listen; history; text; internal format)
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 2n 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 2n-dimensional ball with radius 1/2. - Peter Luschny, Jul 24 2012

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

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.

R. Coquereaux, J.-B. Zuber, Maps, immersions and permutations, arXiv preprint arXiv:1507.03163, 2015. Also J. Knot Theory Ramifications 25, 1650047 (2016), DOI: http://dx.doi.org/10.1142/S0218216516500474

Sylvie Corteel and Lauren Williams, Tableaux Combinatorics for the Asymmetric Exclusion Process II, October 16, 2008. [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, Vol. 5 (2002), Article 02.1.7

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

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.

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. (from Karol A. Penson, Jan 28 2002)

Sum[(-1)^k/(2*k + 1)^n, {k, 0, Infinity}] = (-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]

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,...

...

- Gary W. Adamson, Jul 19 2011

G.f.: 1/(1-4x/(1-4x/(1-8x/(1-8x/(1-12x/(1-12x/(1-16x/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

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

s=1; lst={s}; Do[s+=n*s; AppendTo[lst, s], {n, 3, 5!, 4}]; lst (* Vladimir Joseph Stephan Orlovsky, Nov 08 2008 *)

a[ n_] := With[{m = 2 n}, If[ m < 0, 0, m! SeriesCoefficient[ 1 + Sqrt[Pi] x Exp[x^2] Erf[x], {x, 0, m}]]]; (* Michael Somos, Jan 03 2015 *)

PROG

(PARI)  a(n)=4^n*n!;

(MAGMA) [4^n*Factorial(n): n in [0..20]]; // Vincenzo Librandi, Jul 20 2011

CROSSREFS

Cf. A000142, A007696, A008545, A032031, A000165. a(n)= A051142(n+1, 0) (first column of triangle).

Cf. A087299.

Sequence in context: A051489 A295257 A303049 * A201594 A222412 A007763

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

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Last modified April 20 03:03 EDT 2019. Contains 322294 sequences. (Running on oeis4.)