%I #35 Sep 03 2024 01:42:30
%S 0,0,0,0,16,3552,8332800
%N Number of odd reduced Latin squares of order n.
%H Carolin Hannusch, <a href="https://arato.inf.unideb.hu/hannusch.carolin/evenlatinsquares.txt">Python program for latin squares</a>
%F a(n) + A375141(n) = A000315(n).
%e For n=5 the 16 squares are:
%e [[1, 2, 3, 4, 5],[2, 1, 4, 5, 3],[3, 4, 5, 1, 2],[4, 5, 2, 3, 1],[5, 3, 1, 2, 4]];
%e [[1, 2, 3, 4, 5],[2, 1, 4, 5, 3],[3, 4, 5, 2, 1],[4, 5, 1, 3, 2],[5, 3, 2, 1, 4]];
%e [[1, 2, 3, 4, 5],[2, 1, 5, 3, 4],[3, 5, 4, 1, 2],[4, 3, 2, 5, 1],[5, 4, 1, 2, 3]];
%e [[1, 2, 3, 4, 5],[2, 1, 5, 3, 4],[3, 5, 4, 2, 1],[4, 3, 1, 5, 2],[5, 4, 2, 1, 3]];
%e [[1, 2, 3, 4, 5],[2, 3, 1, 5, 4],[3, 4, 5, 2, 1],[4, 5, 2, 1, 3],[5, 1, 4, 3, 2]];
%e [[1, 2, 3, 4, 5],[2, 3, 1, 5, 4],[3, 5, 4, 1, 2],[4, 1, 5, 2, 3],[5, 4, 2, 3, 1]];
%e [[1, 2, 3, 4, 5],[2, 3, 4, 5, 1],[3, 1, 5, 2, 4],[4, 5, 2, 1, 3],[5, 4, 1, 3, 2]];
%e [[1, 2, 3, 4, 5],[2, 3, 5, 1, 4],[3, 1, 4, 5, 2],[4, 5, 1, 2, 3],[5, 4, 2, 3, 1]];
%e [[1, 2, 3, 4, 5],[2, 4, 1, 5, 3],[3, 5, 2, 1, 4],[4, 1, 5, 3, 2],[5, 3, 4, 2, 1]];
%e [[1, 2, 3, 4, 5],[2, 4, 5, 1, 3],[3, 1, 2, 5, 4],[4, 5, 1, 3, 2],[5, 3, 4, 2, 1]];
%e [[1, 2, 3, 4, 5],[2, 4, 5, 1, 3],[3, 5, 1, 2, 4],[4, 3, 2, 5, 1],[5, 1, 4, 3, 2]];
%e [[1, 2, 3, 4, 5],[2, 4, 5, 3, 1],[3, 5, 1, 2, 4],[4, 1, 2, 5, 3],[5, 3, 4, 1, 2]];
%e [[1, 2, 3, 4, 5],[2, 5, 1, 3, 4],[3, 4, 2, 5, 1],[4, 3, 5, 1, 2],[5, 1, 4, 2, 3]];
%e [[1, 2, 3, 4, 5],[2, 5, 4, 1, 3],[3, 4, 1, 5, 2],[4, 3, 5, 2, 1],[5, 1, 2, 3, 4]];
%e [[1, 2, 3, 4, 5],[2, 5, 4, 3, 1],[3, 1, 2, 5, 4],[4, 3, 5, 1, 2],[5, 4, 1, 2, 3]];
%e [[1, 2, 3, 4, 5],[2, 5, 4, 3, 1],[3, 4, 1, 5, 2],[4, 1, 5, 2, 3],[5, 3, 2, 1, 4]].
%Y Cf. A114629, A375141.
%K nonn,more,hard
%O 1,5
%A _Carolin Hannusch_, Jul 31 2024