|
%I #14 May 25 2023 08:02:42
%S 1,8,15,24,16,27,108,324,774,1620,2268,1584,64,396,1728,7200,27648,
%T 87480,232704,476928,663552,463104,125,1050,6000,35800,198000,977400,
%U 4392000,17068320,56376000,151632000,311040000,430272000,299289600
%N Triangle read by rows T(n,k): number of undirected cycles of length k in the complete tripartite graph K_{n,n,n} (n = 1...; k = 3..3n)
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CyclePolynomial.html">Cycle Polynomial</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteTripartiteGraph.html">Complete Tripartite Graph</a>
%F Row sums of T(n,k) give A234616(n).
%e Written as cycle polynomials:
%e x^3
%e 8 x^3 + 15 x^4 + 24 x^5 + 16 x^6
%e 27 x^3 + 108 x^4 + 324 x^5 + 774 x^6 + 1620 x^7 + 2268 x^8 + 1584 x^9
%e 64 x^3 + 396 x^4 + 1728 x^5 + 7200 x^6 + 27648 x^7 + 87480 x^8 + 232704 x^9 + 476928 x^10 + 663552 x^11 + 463104 x^12
%e giving the array
%e 1
%e 8, 15, 24, 16
%e 27, 108, 324, 774, 1620, 2268, 1584
%e 64, 396, 1728, 7200, 27648, 87480, 232704, 476928, 663552, 463104
%t Table[Tally[Length /@ FindCycle[CompleteGraph[{n, n, n}], Infinity, All]][[All, 2]], {n, 4}] // Flatten
%Y Cf. A234616 (number of undirected cycles in K_{n,n,n}).
%Y Cf. A144151 (cycle polynomial coefficients of complete graph K_n).
%Y Cf. A291909 (cycle polynomial coefficients of complete bipartite graph K_{n,n}).
%K nonn,tabf
%O 1,2
%A _Eric W. Weisstein_, Dec 15 2017
|
