%I #22 Mar 25 2022 14:38:31

%S 1,2,12,48,480,1440,30240,161280,2177280,14515200,399168000,

%T 1916006400,74724249600,523069747200,10461394944000,167382319104000,

%U 5690998849536000,38414242234368000,2189611807358976000,19463216065413120000,613091306060513280000

%N Number of cyclic Latin squares of order n.

%C A cyclic Latin square is a Latin square in which row i is obtained by cyclically shifting row i-1 by d places.

%C Equivalently, a Latin square is cyclic if and only if each row is a cyclic permutation of the first row and each column is a cyclic permutation of the first column.

%H Eduard I. Vatutin, <a href="http://evatutin.narod.ru/evatutin_ls_euler_func.pdf">Enumerating cyclic Latin squares and Euler totient function calculating using them</a>, High-performance computing systems and technologies, 2020, Vol. 4, No. 2, pp. 40-48. (in Russian)

%H Eduard I. Vatutin, <a href="https://vk.com/wall162891802_1427">About the number of cyclic Latin squares and cyclic diagonal Latin squares</a> (in Russian).

%F a(n) = phi(n) * n!.

%F a(n) = A000010(n) * A000142(n).

%e For n=5 there are 4 cyclic Latin squares with the first row in natural order:

%e 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4

%e 1 2 3 4 0 2 3 4 0 1 3 4 0 1 2 4 0 1 2 3

%e 2 3 4 0 1 4 0 1 2 3 1 2 3 4 0 3 4 0 1 2

%e 3 4 0 1 2 1 2 3 4 0 4 0 1 2 3 2 3 4 0 1

%e 4 0 1 2 3 3 4 0 1 2 2 3 4 0 1 1 2 3 4 0

%e and 4*5! = 480 cyclic Latin squares.

%Y Cf. A000010, A000142, A074930, A123565.

%K nonn,easy

%O 1,2

%A _Eduard I. Vatutin_, Nov 01 2020