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A212085 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).

%I A212085 #53 Feb 16 2025 08:33:17
%S A212085 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 A212085 2420,630,42,0,2,378,7332,18500,9750,1302,56,0,2,762,28884,127220,
%U A212085 121590,30702,2408,72,0,2,1530,112740,825860,1324470,583422,81032,4104,90
%N A212085 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 A212085 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 A212085 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 A212085 Alois P. Heinz, <a href="/A212085/b212085.txt">Antidiagonals n = 1..141, flattened</a>
%H A212085 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CompleteBipartiteGraph.html">Complete Bipartite Graph</a>
%H A212085 Wikipedia, <a href="https://en.wikipedia.org/wiki/Chromatic_polynomial">Chromatic Polynomial</a>
%F A212085 A(n,k) = Sum_{j=1..k} (n-j)^k * S2(k,j) * Product_{i=0..j-1} (n-i).
%F A212085 A(n,n)/n = A282245(n).
%e A212085 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 A212085 Square array A(n,k) begins:
%e A212085    0,   0,    0,      0,       0,        0,         0, ...
%e A212085    2,   2,    2,      2,       2,        2,         2, ...
%e A212085    6,  18,   42,     90,     186,      378,       762, ...
%e A212085   12,  84,  420,   1812,    7332,    28884,    112740, ...
%e A212085   20, 260, 2420,  18500,  127220,   825860,   5191220, ...
%e A212085   30, 630, 9750, 121590, 1324470, 13284630, 126657750, ...
%p A212085 A:= (n, k)-> add(Stirling2(k, j) *mul(n-i, i=0..j-1) *(n-j)^k, j=1..k):
%p A212085 seq(seq(A(n, 1+d-n), n=1..d), d=1..12);
%t A212085 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 A212085 Rows n=1-3 give: A000004, A007395, A068293(k+1).
%Y A212085 Columns k=1-2 give: A002378(n-1), A091940.
%Y A212085 Cf. A008277, A212084, A266695, A282245.
%K A212085 nonn,tabl
%O A212085 1,3
%A A212085 _Alois P. Heinz_, Apr 30 2012