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%I #28 Jun 01 2022 07:59:35
%S 0,0,0,0,1,1,0,0,1,2,0,1,2,4,13,0,0,1,2,13,80,0,1,1,9,13,82,579,0,0,2,
%T 2,13,80,579,4738,0,1,1,4,44,82,579,4740,43387,0,0,1,2,13,80,579,4738,
%U 43387,439792,0,1,2,9,13,265,579,4752,43390,439794,4890741
%N Table of generalized ménage numbers read by antidiagonals upward: T(n,k) is the number of permutations pi in S_k such that pi(i) != i, i+n (mod k) for all i; n, k >= 1.
%C The recurrence for the second row comes from Doron Zeilberger's MENAGE program, available via the arXiv reference.
%H D. Zeilberger, <a href="https://arxiv.org/abs/1401.1089">Automatic Enumeration of Generalized Menage Numbers</a>, arXiv preprint arXiv:1401.1089 [math.CO], 2014.
%F T(n,n) = A000166(n) for n >= 1.
%F T(1,k) = A000179(k).
%F T(k-1,k) = A000179(k) for k >= 2.
%F T(n,k) = T(n+k, k).
%F T(2,k) = k*T(2,k-1) + 3*T(2,k-2) + (-2*k+6)*T(2,k-3) - 3*T(2,k-4) + (k-6)*T(2,k-5) + T(2,k-6) for k > 8.
%F T(n,k) = A277256(gcd(n,k),k/gcd(n,k)). - _Pontus von Brömssen_, May 31 2022
%e Table begins:
%e n\k | 1 2 3 4 5 6 7 8
%e ----+--------------------------
%e 1 | 0 0 1 2 13 80 579 4738
%e 2 | 0 1 1 4 13 82 579 4740
%e 3 | 0 0 2 2 13 80 579 4738
%e 4 | 0 1 1 9 13 82 579 4752
%e 5 | 0 0 1 2 44 80 579 4738
%e 6 | 0 1 2 4 13 265 579 4740
%e 7 | 0 0 1 2 13 80 1854 4738
%e 8 | 0 1 1 9 13 82 579 14833
%Y Cf. A000166, A000179, A277256, A354408.
%K nonn,tabl
%O 1,10
%A _Peter Kagey_, Feb 11 2021
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