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1, 4, 32, 384, 6144, 122880, 2949120, 82575360, 2642411520, 95126814720, 3805072588800, 167423193907200, 8036313307545600, 417888291992371200, 23401744351572787200, 1404104661094367232000, 89862698310039502848000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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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.
The partially asymmetric exclusion process (PASEP) is an important model from statistical mechanics which describes a system of interacting particles hopping left and right on a one-dimensional lattice of n sites.
It has been cited as a model for traffic flow and protein synthesis. In its most general form, particles may enter and exit at the left with probabilities alpha and gamma and they may exit and enter at the right with probabilities beta and delta.
In the bulk, the probability of hopping left is q times the probability of hopping right. In previous work we used the matrix ansatz to give a combinatorial formula for the steady state probability of each state of the PASEP, when gamma=delta=0. The formula was the generating function for permutation tableaux of a fixed shape, weighted according to three statistics.
In this paper we give a simple one-parameter generalization of the matrix ansatz, then use it to generalize our results about the PASEP to the case of general alpha, beta, gamma, delta (and q=1). We replace permutation tableaux by the slightly more general bordered permutation tableaux, which we show have cardinality 4^n n!. We also state our results in terms of alternative tableaux. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Oct 17 2008]
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.
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REFERENCES
| A. Hardt and J. M. Troyka, Restricted symmetric signed permutations, pre-print. [For more information, email troykaj(AT)carleton.edu.]
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.
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LINKS
| Vincenzo Librandi, Table of n, a(n) for n = 0..100
Sylvie Corteel and Lauren Williams, Tableaux Combinatorics for the Asymmetric Exclusion Process II, October 16, 2008. [From Jonathan Vos Post (jvospost3(AT)gmail.com), Oct 17 2008]
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
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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, penson(AT)lptl.jussieu.fr, 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 (pbarry(AT)wit.ie), May 04 2007
E.g.f.: With interpolated zeros, 1+sqrt(pi)*x*exp(x^2)*erf(x). [From Paul Barry (pbarry(AT)wit.ie), 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) . - DELEHAM Philippe, Jan 08 2012
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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
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MATHEMATICA
| s=1; lst={s}; Do[s+=n*s; AppendTo[lst, s], {n, 3, 5!, 4}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 08 2008]
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PROG
| (PARI) a(n)=4^n*n!;
(MAGMA) [4^n*Factorial(n): n in [0..20]]; // Vincenzo Librandi, Jul 20 2011
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CROSSREFS
| Cf. A000142, A007696, A008545, A032031, A000165. a(n)= A051142(n+1, 0) (first column of triangle).
Contribution from Johannes W. Meijer (meijgia(AT)hotmail.com), Nov 12 2009: (Start)
Equals the first right hand column of A167557.
Equals the first right hand column of A167569.
(End)
Sequence in context: A137432 A177750 A051489 * A201594 A007763 A195193
Adjacent sequences: A047050 A047051 A047052 * A047054 A047055 A047056
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KEYWORD
| nonn,easy
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AUTHOR
| Joe Keane (jgk(AT)jgk.org)
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EXTENSIONS
| Edited by Karol A. Penson (penson(AT)lptl.jussieu.fr), Jan 22, 2002
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