This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A369923 #15 Feb 21 2025 16:46:56 %S A369923 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 A369923 75843,866,1,0,1,44189,20167651,83002866,4446741,5812,1,0,1,673471, %U A369923 3980871156,158861646466,55279816356,276154969,40048,1,0 %N 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. %C A369923 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 A369923 Andrew Howroyd, <a href="/A369923/b369923.txt">Table of n, a(n) for n = 1..1275</a> (first 51 antidiagonals) %H A369923 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 A369923 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 A369923 Array begins: %e A369923 n\k| 1 2 3 4 5 6 ... %e A369923 ---+----------------------------------------------------------- %e A369923 1 | 0 1 1 1 1 1 ... %e A369923 2 | 0 1 4 31 293 3326 ... %e A369923 3 | 0 1 22 1415 140343 20167651 ... %e A369923 4 | 0 1 134 75843 83002866 158861646466 ... %e A369923 5 | 0 1 866 4446741 55279816356 1450728060971387 ... %e A369923 6 | 0 1 5812 276154969 39738077935264 14571371516350429940 ... %e A369923 ... %t A369923 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 *) %o A369923 (PARI) \\ compare with A322013. %o A369923 q(n, x) = sum(i=1, n, (-1)^(n-i) * binomial(n-1, n-i) * x^i/i!) %o A369923 T(n, k) = if(k > 1, subst(serlaplace(n*q(n, x)^k/x), x, 1)/(k-1)!, 0) %Y A369923 Rows 2..6 are A003436, A348813, A348815, A348818, A348821. %Y A369923 Column 3 is A197657, column 4 appears to be A209183(n)/2. %Y A369923 Cf. A322013 (without linearly adjacent elements equal), A322093. %K A369923 nonn,tabl %O A369923 1,8 %A A369923 _Andrew Howroyd_, Feb 05 2024