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%I #31 Apr 26 2024 07:06:33
%S 2,8,60,816,17520,550080,23839200,1365799680,100053999360,
%T 9127781913600,1015061950425600,135193044668774400,
%U 21248464632595200000,3891825697262043340800,821745573997874093568000
%N Number of full spectrum rook's walks on a (2 X n) board.
%C A rook must land on each square exactly once, but may start and end anywhere and may intersect its own path.
%D Inspired by _Leroy Quet_ in a Jul 05 2004 posting to the Seqfan mailing list.
%H Y. Cha, <a href="http://purl.flvc.org/fsu/fd/FSU_migr_etd-3960">Closed form solutions of difference equations</a> (2011) PhD Thesis, Florida State University, Example 5.1.1
%F D-finite with recurrence: a(n+1) = n*(n+1)*(a(n) + a(n-1)) for n > 1.
%F Further refinement gives: a(n+1) = 2*(n+1)! * Sum_{k=0..floor(n/2)} (P(n-k, k) * C(n-k, k) + P(n-k, k+1) * C(n-1-k, i)), where P(i,j) = i!/j!.
%F Conjecture: a(n) = 2*n!*A102038(n). - _Mikhail Kurkov_, Feb 07 2019
%e Tagging the squares on a (3 X 2) board with A,B,C/D,E,F, the 10 tours starting at A are ABCFDE, ABCFED, ABEDFC, ACBEDF, ACBEFD, ACFDEB, ADEBCF, ADEFCB, ADFCBE, ADFEBC. There are a similar 10 tours starting at each of the other 5 squares, so a(3) = 6 * 10 = 60.
%Y Column 2 of A269565.
%Y Cf. A096970 and references to "rook tours" or "rook walks".
%K nonn,easy,walk,changed
%O 1,1
%A _Hugo van der Sanden_, Jul 22 2004
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