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%I #52 Oct 02 2022 00:36:45
%S 0,0,2,0,2,6,0,2,18,12,0,2,42,84,20,0,2,90,420,260,30,0,2,186,1812,
%T 2420,630,42,0,2,378,7332,18500,9750,1302,56,0,2,762,28884,127220,
%U 121590,30702,2408,72,0,2,1530,112740,825860,1324470,583422,81032,4104,90
%N Square array A(n,k), n>=1, k>=1, read by antidiagonals: A(n,k) is the number of n-colorings of the complete bipartite graph K_(k,k).
%C The complete bipartite graph K_(n,n) has 2*n vertices and n^2 = A000290(n) edges. The chromatic polynomial of K_(n,n) has 2*n+1 coefficients.
%C A(n,k) is the number of pairs of strings of length k over an alphabet of size n such that the strings do not share any letter. - _Lin Zhangruiyu_, Aug 19 2022
%H Alois P. Heinz, <a href="/A212085/b212085.txt">Antidiagonals n = 1..141, flattened</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Chromatic_polynomial">Chromatic Polynomial</a>
%F A(n,k) = Sum_{j=1..k} (n-j)^k * S2(k,j) * Product_{i=0..j-1} (n-i).
%F A(n,n)/n = A282245(n).
%e A(3,1) = 6 because there are 6 3-colorings of the complete bipartite graph K_(1,1): 1-2, 1-3, 2-1, 2-3, 3-1, 3-2.
%e Square array A(n,k) begins:
%e 0, 0, 0, 0, 0, 0, 0, ...
%e 2, 2, 2, 2, 2, 2, 2, ...
%e 6, 18, 42, 90, 186, 378, 762, ...
%e 12, 84, 420, 1812, 7332, 28884, 112740, ...
%e 20, 260, 2420, 18500, 127220, 825860, 5191220, ...
%e 30, 630, 9750, 121590, 1324470, 13284630, 126657750, ...
%p A:= (n, k)-> add(Stirling2(k, j) *mul(n-i, i=0..j-1) *(n-j)^k, j=1..k):
%p seq(seq(A(n, 1+d-n), n=1..d), d=1..12);
%t a[n_, k_] := Sum[(-1)^j*(n-j)^k*Pochhammer[-n, j]*StirlingS2[k, j], {j, 1, k}]; Table[a[n-k, k], {n, 1, 11}, {k, n-1, 1, -1}] // Flatten (* _Jean-François Alcover_, Dec 11 2013 *)
%Y Rows n=1-3 give: A000004, A007395, A068293(k+1).
%Y Columns k=1-2 give: A002378(n-1), A091940.
%Y Cf. A008277, A212084, A266695, A282245.
%K nonn,tabl
%O 1,3
%A _Alois P. Heinz_, Apr 30 2012
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