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A144045 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). 2
1, 6, 222, 9918, 486924, 25267236, 1359631776, 75059524392, 4223303759148, 241144782230124, 13930829740017132, 812470444305924300, 47760356825349969600, 2826309951801018736800, 168207011284961649886800, 10060178088232285063542768, 604273284101165691102038556 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Alin Bostan, Calcul Formel pour la Combinatoire des Marches [The text is in English], Habilitation à Diriger des Recherches, Laboratoire d’Informatique de Paris Nord, Université Paris 13, December 2017.
M. Kauers and D. Zeilberger, The Computational Challenge of Enumerating High-Dimensional Rook Walks, arXiv:1011.4671 [math.CO], 2010.
FORMULA
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.
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]
Asymptotics: a(n) ~ 9*sqrt(3)/(40*Pi*n)*64^(n-1). - Frederic Chyzak, 2010
EXAMPLE
a(2)=6 because there are 6 Rook paths from (1,1,1) to (2,2,2).
G.f. = x + 6*x^2 + 222*x^3 + 9918*x^4 + 486924*x^5 + 25267236*x^6 + ...
CROSSREFS
Cf. A051708.
Row d=3 of A181731.
Sequence in context: A331477 A144658 A203637 * A355669 A061610 A054324
KEYWORD
nonn
AUTHOR
Martin J. Erickson (erickson(AT)truman.edu), Sep 08 2008
STATUS
approved

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Last modified April 15 04:32 EDT 2024. Contains 371667 sequences. (Running on oeis4.)