0,2

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]

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.

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

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]

a(n) = 4^n*n!. E.g.f.: (1-4*x)^-1; 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

restart: G(x):=(1-4*x)^(n-2): f[0]:=G(x): for n from 1 to 29 do f[n]:=diff(f[n-1], x) od:x:=0:seq(f[n], n=0..16); # [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Apr 04 2009]

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]

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: A002005 A123309 A051489 this_sequence A007763 A005263 A113131

Adjacent sequences: A047050 A047051 A047052 this_sequence A047054 A047055 A047056

nonn,easy

Joe Keane (jgk(AT)jgk.org)

Edited by Karol A. Penson (penson(AT)lptl.jussieu.fr), Jan 22, 2002