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A291774 Triangle read by rows: chromatic invariant T(n,m) of the complete bipartite graph K_{m,n}.

%I A291774 #13 Feb 16 2025 08:33:51
%S A291774 1,0,1,0,1,5,0,1,13,73,0,1,29,301,2069,0,1,61,1081,11581,95401,0,1,
%T A291774 125,3613,57749,673261,6487445,0,1,253,11593,268381,4306681,55213453,
%U A291774 610093513,0,1,509,36301,1191989,25794781,431525429,6077248381,75796724309,0,1,1021,111961,5136061,147587401,3173843821,56153444761,864806272861,12020754177001
%N A291774 Triangle read by rows: chromatic invariant T(n,m) of the complete bipartite graph K_{m,n}.
%H A291774 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/ChromaticInvariant.html">Chromatic Invariant</a>
%H A291774 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>
%F A291774 T(m,n) = Sum_{k = 0..m-1} k!*(-1)^(k + m)*(k + 1)^n*Stirling2(m, k + 2) for max(m,n) > 1.
%e A291774 Triangle begins:
%e A291774 1
%e A291774 0 1
%e A291774 0 1 5
%e A291774 0 1 13 73
%e A291774 0 1 29 301 2069
%t A291774 Join[{1}, Table[Sum[k! (-1)^(k + m) (k + 1)^n StirlingS2[m, k + 2], {k, 0, m - 1}], {n, 2, 10}, {m, n}]] // Flatten
%Y A291774 Main diagonal gives A048144.
%K A291774 nonn,tabl
%O A291774 1,6
%A A291774 _Eric W. Weisstein_, Aug 31 2017