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%I #11 Feb 06 2024 00:49:29
%S 0,1,0,1,1,0,1,4,1,0,1,31,22,1,0,1,293,1415,134,1,0,1,3326,140343,
%T 75843,866,1,0,1,44189,20167651,83002866,4446741,5812,1,0,1,673471,
%U 3980871156,158861646466,55279816356,276154969,40048,1,0
%N Array read by antidiagonals: A(n,k) is the number of permutations of n copies of 1..k with values introduced in order and without cyclically adjacent elements equal.
%C Also, T(n,k) is the number of generalized chord labeled loopless diagrams with k parts of K_n. See the Krasko reference for a full definition.
%H Andrew Howroyd, <a href="/A369923/b369923.txt">Table of n, a(n) for n = 1..1275</a> (first 51 antidiagonals)
%H Evgeniy Krasko, Igor Labutin, and Alexander Omelchenko, <a href="https://arxiv.org/abs/1709.03218">Enumeration of Labelled and Unlabelled Hamiltonian Cycles in Complete k-partite Graphs</a>, arXiv:1709.03218 [math.CO], 2017.
%H Mathematics.StackExchange, <a href="https://math.stackexchange.com/questions/129451/find-the-number-of-arrangements-of-k-mbox-1s-k-mbox-2s-cdots">Find the number of k 1's, k 2's, ... , k n's - total kn cards</a>, Apr 08 2012.
%e Array begins:
%e n\k| 1 2 3 4 5 6 ...
%e ---+-----------------------------------------------------------
%e 1 | 0 1 1 1 1 1 ...
%e 2 | 0 1 4 31 293 3326 ...
%e 3 | 0 1 22 1415 140343 20167651 ...
%e 4 | 0 1 134 75843 83002866 158861646466 ...
%e 5 | 0 1 866 4446741 55279816356 1450728060971387 ...
%e 6 | 0 1 5812 276154969 39738077935264 14571371516350429940 ...
%e ...
%o (PARI) \\ compare with A322013.
%o q(n, x) = sum(i=1, n, (-1)^(n-i) * binomial(n-1, n-i) * x^i/i!)
%o T(n, k) = if(k > 1, subst(serlaplace(n*q(n, x)^k/x), x, 1)/(k-1)!, 0)
%Y Rows 2..6 are A003436, A348813, A348815, A348818, A348821.
%Y Column 3 is A197657, column 4 appears to be A209183(n)/2.
%Y Cf. A322013 (without linearly adjacent elements equal), A322093.
%K nonn,tabl
%O 1,8
%A _Andrew Howroyd_, Feb 05 2024
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