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A384980 Number of proper vertex colorings of the n-complete bipartite graph using exactly 4 interchangeable colors.

%I A384980 #30 Jun 25 2025 00:43:22
%S A384980 0,1,11,61,275,1141,4571,18061,71075,279781,1103531,4363261,17292275,
%T A384980 68670421,273152891,1087959661,4337751875,17308485061,69105848651,
%U A384980 276038071261,1102994217875,4408498475701,17623550326811,70462853802061,281757339138275,1126747061234341,4506141224763371
%N A384980 Number of proper vertex colorings of the n-complete bipartite graph using exactly 4 interchangeable colors.
%C A384980 The complete bipartite graph K(n,n) has 2n vertices partitioned into two sets of size n each, with edges between every pair of vertices from different sets. a(n) counts the number of proper vertex colorings using exactly 4 colors; these are also the number of 4 partitions of the vertices of the 2n vertices. a(n) = 0 for n < 2.
%H A384980 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>.
%H A384980 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (10,-35,50,-24).
%F A384980 a(n) = Sum_{j = 1..3} Stirling2(n, j) * Stirling2(n, 4-j).
%F A384980 a(n)= (4^n)/4 + (3^n)/3 - 2^(n + 1) + 2.
%F A384980 G.f.: (1/4) * 1/(1 - 4*x) + (1/3) * 1/(1 - 3*x) - 2 * 1/(1 - 2*x) + 2 / (1 - x).
%F A384980 From _Stefano Spezia_, Jun 14 2025: (Start)
%F A384980 a(n) = 2 - 2^(n+2) + 3^n + 4^n.
%F A384980 E.g.f.: exp(x)*(2 - 4*exp(x) + exp(2*x) + exp(3*x)). (End)
%e A384980 a(3) = 11 because K(3,3) has vertices {a1,a2,a3,b1,b2,b3} and there are 11 ways to color properly 6 vertices using all 4 colors.
%t A384980 Table[2*StirlingS2[n, 3] + StirlingS2[n, 2]^2, {n, 1, 50}]
%o A384980 (PARI) a(n) = 2*stirling(n,3,2) + stirling(n,2,2)^2
%Y A384980 Column 4 of A384968.
%Y A384980 Cf. A384981.
%K A384980 nonn,easy
%O A384980 1,3
%A A384980 _Julian Allagan_, Jun 14 2025