A369923 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.
0, 1, 0, 1, 1, 0, 1, 4, 1, 0, 1, 31, 22, 1, 0, 1, 293, 1415, 134, 1, 0, 1, 3326, 140343, 75843, 866, 1, 0, 1, 44189, 20167651, 83002866, 4446741, 5812, 1, 0, 1, 673471, 3980871156, 158861646466, 55279816356, 276154969, 40048, 1, 0
Offset: 1
Examples
Array begins: n\k| 1 2 3 4 5 6 ... ---+----------------------------------------------------------- 1 | 0 1 1 1 1 1 ... 2 | 0 1 4 31 293 3326 ... 3 | 0 1 22 1415 140343 20167651 ... 4 | 0 1 134 75843 83002866 158861646466 ... 5 | 0 1 866 4446741 55279816356 1450728060971387 ... 6 | 0 1 5812 276154969 39738077935264 14571371516350429940 ... ...
Links
- Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 51 antidiagonals)
- Evgeniy Krasko, Igor Labutin, and Alexander Omelchenko, Enumeration of Labelled and Unlabelled Hamiltonian Cycles in Complete k-partite Graphs, arXiv:1709.03218 [math.CO], 2017.
- Mathematics.StackExchange, Find the number of k 1's, k 2's, ... , k n's - total kn cards, Apr 08 2012.
Crossrefs
Programs
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Mathematica
T[n_, k_] := If[k == 1, 0, Expand[(-1)^(k (n + 1))/(k - 1)! n Hypergeometric1F1[1 - n, 2, x]^k x^(k - 1)] /. x^p_ :> p!] (* Eric W. Weisstein, Feb 20 2025 *)
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PARI
\\ compare with A322013. q(n, x) = sum(i=1, n, (-1)^(n-i) * binomial(n-1, n-i) * x^i/i!) T(n, k) = if(k > 1, subst(serlaplace(n*q(n, x)^k/x), x, 1)/(k-1)!, 0)
Comments