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%I #21 May 10 2020 08:37:11
%S 1,6,222,9918,486924,25267236,1359631776,75059524392,4223303759148,
%T 241144782230124,13930829740017132,812470444305924300,
%U 47760356825349969600,2826309951801018736800,168207011284961649886800,10060178088232285063542768,604273284101165691102038556
%N Number of paths of a chess Rook in a cube, from (1,1,1) to (n,n,n), where the rook may move in steps that are multiples of (1,0,0), (0,0,1), or (0,0,1).
%H Alin Bostan, <a href="https://specfun.inria.fr/bostan/HDR.pdf">Calcul Formel pour la Combinatoire des Marches</a> [The text is in English], Habilitation à Diriger des Recherches, Laboratoire d’Informatique de Paris Nord, Université Paris 13, December 2017.
%H M. Kauers and D. Zeilberger, <a href="http://arXiv.org/abs/1011.4671">The Computational Challenge of Enumerating High-Dimensional Rook Walks</a>, arXiv:1011.4671 [math.CO], 2010.
%F a(n) satisfies the recurrence relation a(1) = 1; a(2) = 6; a(3) = 222; a(4) = 9918; a(n) = ((-121 n^3 + 575n^2 - 872n + 412)a(n - 1) + (-475n^3 + 4887n^2 - 16202n + 17448)a(n - 2) + (1746n^3 - 19818n^2 + 75060n - 94896)a(n - 3) + (-1152n^3 + 16128n^2 - 74880n + 115200)a(n - 4))/(-(2n^3 - 8n^2 + 10n - 4)), n>= 5.
%F G.f.: 1+int(6*hypergeom([1/3, 2/3],[2],27*x*(3*x-2)/(4*x-1)^3)/((4*x-1)*(64*x-1)),x). [_Mark van Hoeij_, Dec 10 2009]
%F Asymptotics: a(n) ~ 9*sqrt(3)/(40*Pi*n)*64^(n-1). - Frederic Chyzak, 2010
%e a(2)=6 because there are 6 Rook paths from (1,1,1) to (2,2,2).
%e G.f. = x + 6*x^2 + 222*x^3 + 9918*x^4 + 486924*x^5 + 25267236*x^6 + ...
%Y Cf. A051708.
%Y Row d=3 of A181731.
%K nonn
%O 1,2
%A Martin J. Erickson (erickson(AT)truman.edu), Sep 08 2008
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