A092616 -id:A092616 - cp's OEIS frontend

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

Joe Keane (jgk(AT)jgk.org)

Original name was "Quadruple factorial numbers".
For n >= 1, a(n) is the order of the wreath product of the cyclic group C_4 and the symmetric group S_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

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

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.
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.

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

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

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

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

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 *)

PARI
```
a(n)=4^n*n!;
```

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: a(n) = Integral_{x=0..oo} x^n*exp(-x/4)/4, n >= 0. This representation is unique. - Karol A. Penson, Jan 28 2002 [corrected by Jason Yuen, May 04 2025]
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)

Edited by Karol A. Penson, Jan 22 2002

A029571 Number of permutations of an n-set containing a 4-cycle.

Original entry on oeis.org

0, 0, 0, 0, 6, 30, 180, 1260, 8820, 79380, 793800, 8731800, 106029000, 1378377000, 19297278000, 289459170000, 4627941318000, 78675002406000, 1416150043308000, 26906850822852000, 538156815464268000

Offset: 0

Rob Pratt

nonn

Robert Israel, Table of n, a(n) for n = 0..449

Column k=4 of A293211.

L:= [seq( 1 - add((-1)^k/(k!*4^k),k=0..m),m=0..10)]:
seq(seq((4*m+j)!*L[m+1],j=0..3),m=0..10); # Robert Israel, Dec 07 2016

a[n_] := n! (1 - Sum[(-1)^k/(k! 4^k), {k, 0, Floor[n/4]}]);
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 19 2019 *)

a(n)=n! * (1 - sum(k=0,floor(n/4), (-1)^k/(k!*4^k) ) ); \\ Joerg Arndt, Aug 08 2013

a(n) = n! * (1 - sum(k=0..floor(n/4), (-1)^k/(k!*4^k) ) ).
a(n)/n! is asymptotic to 1-e^(-1/4) = 1 - A092616.
a(n) = n! (1 - Gamma(floor(n/4)+1,-1/4)*exp(1/4)/(floor(n/4))!). - Robert Israel, Dec 07 2016
E.g.f.: (1-exp(-x^4/4))/(1-x). - Alois P. Heinz, Oct 11 2017

A113550 a(n) = product of n successive numbers up to n, if n is odd a(n) = n(n-1).. = n!, if n is even a(n) = n(n+1)(n+2)... 'n' terms.

Original entry on oeis.org

1, 6, 6, 840, 120, 332640, 5040, 259459200, 362880, 335221286400, 39916800, 647647525324800, 6227020800, 1748648318376960000, 1307674368000, 6288139352883548160000, 355687428096000, 29051203810321992499200000, 121645100408832000, 167683548393178540705382400000

Offset: 1

Amarnath Murthy, Nov 03 2005

			a(3) = 3*2*1 = 6.
a(4) = 4*5*6*7 = 840.

G. C. Greubel, Table of n, a(n) for n = 1..350

Cf. A113549.

Cf. A019704, A073742, A092042, A092616, A333419, A367563.

n = 1; anfunc[n_] := (If [EvenQ[n], {an = n, Do[an = an*(n + i), {i, n - 1}]}, an = n! ]; an); Table[anfunc[n], {n, 1, 20}] (* Elizabeth A. Blickley (Elizabeth.Blickley(AT)gmail.com), Mar 10 2006 *)

a(2n-1) = (2n-1)!, a(2n) = (4n-1)!/(2n-1)!.
a(2n-1)*a(2n) = (4n-1)!.
Sum_{n>=1} 1/a(n) = sinh(1) + (sqrt(Pi)/2) * (exp(1/4) * erf(1/2) - exp(-1/4) * erfi(1/2)). - Amiram Eldar, Aug 15 2025

More terms from Elizabeth A. Blickley (Elizabeth.Blickley(AT)gmail.com), Mar 10 2006

cp's OEIS Frontend

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

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A029571 Number of permutations of an n-set containing a 4-cycle.

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A113550 a(n) = product of n successive numbers up to n, if n is odd a(n) = n(n-1).. = n!, if n is even a(n) = n(n+1)(n+2)... 'n' terms.

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A029571 Number of permutations of an n-set containing a 4-cycle.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A113550 a(n) = product of n successive numbers up to n, if n is odd a(n) = n*(n-1)*.. = n!, if n is even a(n) = n(n+1)(n+2)... 'n' terms.

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Author

Keywords

Examples

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Mathematica

Formula

Extensions

A113550 a(n) = product of n successive numbers up to n, if n is odd a(n) = n(n-1).. = n!, if n is even a(n) = n(n+1)(n+2)... 'n' terms.