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%I #30 Aug 08 2023 22:22:35
%S 1,0,0,48,960,0,14192640,5449973760,118753937326080
%N Number of centrally symmetric diagonal Latin squares of order n.
%C Centrally symmetric diagon Latin square is a square with one-to-one correspondence between elements within all pairs a[i][j] and a[n-1-i][n-1-j] (numbering of rows and columns from 0 to n-1).
%C It seems that a(n)=0 for n=2 and n=3 (diagonal Latin squares of these sizes don't exist) and for n=2 (mod 4).
%C Every doubly symmetric diagonal Latin square also has central symmetry. The converse is not true in general. It follows that A292517(n) <= a(4n). - _Eduard I. Vatutin_, May 26 2021
%H E. I. Vatutin, <a href="http://forum.boinc.ru/default.aspx?g=posts&m=89455#post89455">Discussion about properties of diagonal Latin squares at forum.boinc.ru</a> (in Russian)
%H E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, M. O. Manzuk, N. N. Nikitina, V. S. Titov, <a href="http://evatutin.narod.ru/evatutin_co_dls_centr_symm.pdf">Properties of central symmetry for diagonal Latin squares</a>, High-performance computing systems and technologies, No. 1 (8), 2018, pp. 74-78. (in Russian)
%H E. I. Vatutin, S. E. Kochemazov, O. S. Zaikin, M. O. Manzuk, N. N. Nikitina, V. S. Titov, <a href="https://jpit.az/uploads/article/az/2019_2/CENTRAL_SYMMETRY_PROPERTIES_FOR_DIAGONAL_LATIN_SQUARES.pdf">Central Symmetry Properties for Diagonal Latin Squares</a>, Problems of Information Technology, No. 2, 2019, pp. 3-8. doi: 10.25045/jpit.v10.i2.01.
%H E. I. Vatutin, <a href="http://evatutin.narod.ru/evatutin_dls_spec_types_list.pdf">Special types of diagonal Latin squares</a>, Cloud and distributed computing systems in electronic control conference, within the National supercomputing forum (NSCF - 2022). Pereslavl-Zalessky, 2023. pp. 9-18. (in Russian)
%H Eduard I. Vatutin, <a href="https://vk.com/wall162891802_1635">On the interconnection between double and central symmetries in diagonal Latin squares</a> (in Russian).
%H <a href="/index/La#Latin">Index entries for sequences related to Latin squares and rectangles</a>.
%F a(n) = A293777(n) * n!.
%e 0 1 2 3 4 5 6 7 8
%e 6 3 0 2 7 8 1 4 5
%e 3 2 1 8 6 7 0 5 4
%e 7 8 6 5 1 3 4 0 2
%e 8 6 4 7 2 0 5 3 1
%e 2 7 5 6 8 4 3 1 0
%e 5 4 7 0 3 1 8 2 6
%e 4 5 8 1 0 2 7 6 3
%e 1 0 3 4 5 6 2 8 7
%Y Cf. A292516, A292517, A293777, A340545.
%K nonn,more,hard
%O 1,4
%A _Eduard I. Vatutin_, Oct 16 2017