%I #10 Jan 17 2013 12:39:27
%S 2,12,4,432,72,8,61344,3888,288,16,32866560,665280,21600,960,32,
%T 68307743232,407306880,4328640,95040,2880,64,561981464819712,
%U 965518299648,2948037120,21893760,362880,8064,128
%N Triangular array T(n,k) read by rows: number of labeled relations of the n-set with exactly k connected components.
%C T(n,k) is the number of binary relations R on {1,2,...,n} such that the reflexive, symmetric and transitive closure of R is an equivalence relation with exactly k classes.
%C Row sums are A002416 = 2^(n^2).
%C Column 1 is A062738.
%C T(n,n) = 2^n is the number of binary relations that are a subset of the diagonal relation.
%F E.g.f. for column k: log(A(x))^k/k! where A(x) is the E.g.f. for A002416
%e 2
%e 12 4
%e 432 72 8
%e 61344 3888 288 16
%e 32866560 665280 21600 960 32
%t a=Sum[2^(n^2) x^n/n!,{n,0,10}];
%t Transpose[Table[Drop[Range[0, 10]! CoefficientList[Series[Log[a]^n/n!, {x, 0, 10}],x],1], {n, 1, 10}]] // Grid
%K nonn,tabl
%O 1,1
%A _Geoffrey Critzer_, May 28 2011