|
%I #35 Aug 09 2020 01:17:09
%S 1,14,133,3300,93889,3391086,148674191,7796637196,480640583751,
%T 34370030511334,2818294139246649,262403744798653716,
%U 27506121212584723373,3222018028986227724702,418998630100386520363619,60138044879434564251209580,9477043948863636836099726259,1632099068624734991723488992214
%N Number of 3 X n Latin rectangles such that every element of the second row has the same cyclic order (see comment).
%C We say that an element alpha_i of a permutation alpha of {1,2,...,n} has cyclic order k if it belongs to a cycle of length k of alpha. If every cycle of alpha has length k, then k|n.
%D V. S. Shevelev, Reduced Latin rectangles and square matrices with equal row and column sums, Diskr. Mat. [Journal published by the Academy of Sciences of Russia], 4 (1992), 91-110.
%D V. S. Shevelev, Modern enumeration theory of permutations with restricted positions, Diskr. Mat., 1993, 5, no.1, 3-35 (Russian) [English translation in Discrete Math. and Appl., 1993, 3:3, 229-263 (pp. 255-257)].
%F Let G_n = A000296(n) = n! * Sum_{2*k_2+...+n*k_n=n, k_i>=0} Product_{i=2,...,n} (k_i!*i!^k_i)^(-1). Then a(n) = Sum_{k=0,...,floor(n/2)} binomial(n,k) * G_k * G_(n-k) * u_(n-2*k), where u(n) = A000179(n). - _Vladimir Shevelev_, Mar 30 2016
%Y Cf. A000179, A000186, A000296, A094047, A174556, A174560, A174561.
%K nonn
%O 3,2
%A _Vladimir Shevelev_, Mar 22 2010
%E More terms from _William P. Orrick_, Jul 25 2020
|
