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A384981 Number of proper vertex colorings of the n-complete bipartite graph using exactly 5 interchangeable colors.

%I A384981 #27 Jun 25 2025 00:43:59
%S A384981 0,0,6,86,770,5710,38626,248766,1558290,9603470,58604546,355460446,
%T A384981 2147773810,12945690030,77907271266,468366848126,2813865797330,
%U A384981 16897768573390,101444650414786,608899287739806,3654318951308850,21929599650541550,131592320786851106,789612753560503486
%N A384981 Number of proper vertex colorings of the n-complete bipartite graph using exactly 5 interchangeable colors.
%C A384981 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) = 0 for n < 3 because K(n,n) with n < 3 cannot be partitioned into 5 nonempty independent sets. a(n) counts ways to create exactly 3 additional independent sets beyond the original 2-partite sets by splitting some of the 2-partite sets.
%H A384981 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>.
%H A384981 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (16,-95,260,-324,144).
%F A384981 a(n) = Sum_{j = 1..4} Stirling2(n, j) * Stirling2(n, 5-j).
%F A384981 a(n) = 6^(n - 1) - (5/3)*2^(2*n - 2) - 2*3^(n - 1) + 2^(n + 1) - 4/3 for n >= 1.
%F A384981 G.f.: x*(1/(1 - 6*x) - (5/3)/(1 - 4*x) - 2/(1 - 3*x) + 4/(1 - 2*x) - (4/3)/(1 - x)).
%F A384981 E.g.f.: (exp(x) - 1)^3*(2*exp(3*x) + 6*exp(2*x) + 7*exp(x) - 3)/12. - _Stefano Spezia_, Jun 15 2025
%e A384981 a(3) = 6 because K(3,3) can be partitioned into 5 nonempty independent sets in exactly 6 ways.
%t A384981 Table[2StirlingS2[n, 4] + 2StirlingS2[n, 3]StirlingS2[n, 2], {n, 1, 30}]
%Y A384981 Column 5 of A384968.
%Y A384981 Cf. A008277, A384980.
%K A384981 nonn,easy
%O A384981 1,3
%A A384981 _Julian Allagan_, Jun 14 2025