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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).
5
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, 2420, 630, 42, 0, 2, 378, 7332, 18500, 9750, 1302, 56, 0, 2, 762, 28884, 127220, 121590, 30702, 2408, 72, 0, 2, 1530, 112740, 825860, 1324470, 583422, 81032, 4104, 90
OFFSET
1,3
COMMENTS
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.
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
LINKS
Eric Weisstein's World of Mathematics, Complete Bipartite Graph
FORMULA
A(n,k) = Sum_{j=1..k} (n-j)^k * S2(k,j) * Product_{i=0..j-1} (n-i).
A(n,n)/n = A282245(n).
EXAMPLE
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.
Square array A(n,k) begins:
0, 0, 0, 0, 0, 0, 0, ...
2, 2, 2, 2, 2, 2, 2, ...
6, 18, 42, 90, 186, 378, 762, ...
12, 84, 420, 1812, 7332, 28884, 112740, ...
20, 260, 2420, 18500, 127220, 825860, 5191220, ...
30, 630, 9750, 121590, 1324470, 13284630, 126657750, ...
MAPLE
A:= (n, k)-> add(Stirling2(k, j) *mul(n-i, i=0..j-1) *(n-j)^k, j=1..k):
seq(seq(A(n, 1+d-n), n=1..d), d=1..12);
MATHEMATICA
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 *)
CROSSREFS
Rows n=1-3 give: A000004, A007395, A068293(k+1).
Columns k=1-2 give: A002378(n-1), A091940.
Sequence in context: A033733 A115951 A264954 * A265882 A324253 A208385
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Apr 30 2012
STATUS
approved