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A385432 Triangle read by rows: T(n,k) is the number of proper vertex colorings of the n-complete tripartite graph using exactly k interchangeable colors, 3 <= k <= 3*n.

%I A385432 #8 Jul 03 2025 18:39:52
%S A385432 1,1,3,3,1,1,9,30,45,30,9,1,1,21,165,598,1032,939,471,129,18,1,1,45,
%T A385432 750,5655,19653,36465,39250,25560,10278,2545,375,30,1,1,93,3153,46726,
%U A385432 295905,978588,1881306,2232798,1704405,858530,288768,64743,9495,870,45,1,1,189,12810,364875,3988530,21976122,69388462,134794821,1
%N A385432 Triangle read by rows: T(n,k) is the number of proper vertex colorings of the n-complete tripartite graph using exactly k interchangeable colors, 3 <= k <= 3*n.
%C A385432 Permuting the colors does not change the coloring. T(n,k) is the number of ways to partition the vertices of the n-complete tripartite graph into k independent sets.
%H A385432 Richard P. Stanley, <a href="https://math.mit.edu/~rstan/ec/">Enumerative Combinatorics</a>, Cambridge University Press.
%H A385432 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteMultipartiteGraph.html">Complete Multipartite Graph</a>.
%H A385432 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/StirlingNumberoftheSecondKind.html">Stirling Number of the Second Kind</a>.
%F A385432 T(n,k)=Sum[StirlingS2[n, j1] * StirlingS2[n, j2] * StirlingS2[n, k - j1 - j2],{j1, 1, k - 2}, {j2, 1, k - j1 - 1}]
%e A385432 Triangle begins (n >= 1, k >= 3):
%e A385432   n = 1: [1]
%e A385432   n = 2: [1, 3, 3, 1]
%e A385432   n = 3: [1, 9, 30, 45, 30, 9, 1]
%e A385432   n = 4: [1, 21, 165, 598, 1032, 939, 471, 129, 18, 1]
%e A385432   n = 5: [1, 45, 750, 5655, 19653, 36465, 39250, 25560, 10278, 2545, 375, 30, 1]
%e A385432   n = 6: [1, 93, 3153, 46726, 295905, 978588, 1881306, 2232798, 1704405, 858530, 288768, 64743, 9495, 870, 45, 1]...
%o A385432 (PARI) T(n, k) = sum(j1=1, k-2, sum(j2=1, k-j1-1, my(j3 = k - j1 - j2); if(j3 >= 1, stirling(n, j1, 2)*stirling(n, j2, 2)*stirling(n, j3, 2))));
%o A385432 for(n=1, 8, print(vector(3*n - 2, k, T(n, k + 2))))
%Y A385432 Column k=3 is A384988.
%K A385432 nonn,tabf,easy
%O A385432 1,3
%A A385432 _Julian Allagan_, Jun 28 2025