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Number of simple allowable sequences on 1..n.
3

%I #30 Jul 13 2017 10:47:55

%S 1,1,2,32,4608,7028736,132089118720,34998332896051200,

%T 147462169661142781132800,11008782516353752266715850342400,

%U 16061608070479103314001351327405309952000,500842967990688435516159987675099451681186775040000

%N Number of simple allowable sequences on 1..n.

%D J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, CRC Press, 1997, p. 102.

%D G. Kreweras, Sur un problème de scrutin à plus de deux candidats, Publications de l'Institut de Statistique de l'Université de Paris, 26 (1981), 69-87.

%H Alois P. Heinz, <a href="/A018241/b018241.txt">Table of n, a(n) for n = 1..40</a>

%H G. Kreweras, <a href="/A005118/a005118.pdf">Sur un problème de scrutin à plus de deux candidats</a>, Publications de l'Institut de Statistique de l'Université de Paris, 26 (1981), 69-87. [Annotated scanned copy]

%H R. P. Stanley, <a href="http://dx.doi.org/10.1016/S0195-6698(84)80039-6">On the number of reduced decompositions of elements of Coxeter groups</a>, European J. Combin., 5 (1984), 359-372.

%F a(n) = (n-2)!*C(n,2)! / (1^{n-1} * 3^{n-2} * ... * (2n-3)^1).

%F a(n) ~ Pi * exp(n^2/4 - 3*n/2 + 7/24) * n^(n^2/2 + n/2 - 13/24) / (A^(1/2) * 2^(n^2 - n/2 - 19/24)), where A = 1.2824271291... is the Glaisher-Kinkelin constant (see A074962). - _Vaclav Kotesovec_, Nov 13 2014

%p A018241 := proc(n) local i; (n-2)!*binomial(n,2)!/product( (2*i+1)^(n-i-1), i=0..n-2 ); end;

%t a[n_] := (n*(n-1)/2)!*(n-2)! / Product[ (2i+1)^(n-i-1), {i, 0, n-2}]; a[1] = 1; Table[ a[n], {n, 1, 11}] (* _Jean-François Alcover_, Jan 25 2012 *)

%Y Cf. A003121, A005118, A074962.

%K nonn,easy,nice

%O 1,3

%A _N. J. A. Sloane_