%I #39 Apr 15 2023 17:25:09
%S 1,1,0,0,4,4,80,80,4752,4752,440192,440192,59245120,59245120,
%T 10930514688,10930514688,2649865335040,2649865335040,817154768973824,
%U 817154768973824,312426715251262464,312426715251262464,145060238642780180480,145060238642780180480
%N Number of X-based filling of diagonals in a diagonal Latin square of order n with the main diagonal in ascending order.
%C Used for getting strong canonical forms (SCFs) of the diagonal Latin squares and for fast enumerating of the diagonal Latin squares based on equivalence classes.
%C For all t > 0, a(2*t) = a(2*t+1).
%H S. Kochemazov, O. Zaikin, E. Vatutin, and A. Belyshev, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL23/Zaikin/zaikin3.html">Enumerating Diagonal Latin Squares of Order Up to 9</a>, Journal of Integer Sequences. Vol. 23. Iss. 1. 2020. Article 20.1.2.
%H E. I. Vatutin, <a href="https://vk.com/wall162891802_1291">About the number of X-based fillings of diagonals in a diagonal Latin squares of orders 1-15</a> (in Russian).
%H E. I. Vatutin, <a href="https://vk.com/wall162891802_1293">About the a(2*t)=a(2*t+1) equality</a> (in Russian).
%H E. I. Vatutin, A. D. Belyshev, N. N. Nikitina, and M. O. Manzuk, <a href="http://evatutin.narod.ru/evatutin_dls_scf_gen.pdf">Use of X-based diagonal fillings and ESODLS CMS schemes for enumeration of main classes of diagonal Latin squares</a>, Telecommunications, 2023, No. 1, pp. 2-16, DOI: 10.31044/1684-2588-2023-0-1-2-16 (in Russian).
%H <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>.
%F a(n) = A337303(n)/n!.
%F a(n) = A000316(floor(n/2)). - _Andrew Howroyd_ and _Eduard I. Vatutin_, Oct 08 2020
%e For n=4 there are 4 different X-based fillings of diagonals with main diagonal fixed to [0 1 2 3]:
%e 0 . . 1 0 . . 1 0 . . 2 0 . . 2
%e . 1 0 . . 1 3 . . 1 0 . . 1 3 .
%e . 3 2 . . 0 2 . . 3 2 . . 0 2 .
%e 2 . . 3 2 . . 3 1 . . 3 1 . . 3
%Y Cf. A000316, A309283, A274171, A337303.
%K nonn
%O 0,5
%A _Eduard I. Vatutin_, Aug 22 2020
%E More terms from _Alois P. Heinz_, Oct 08 2020
%E a(0)=1 prepended by _Andrew Howroyd_, Oct 09 2020