OFFSET
1,1
COMMENTS
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.
Row sums are A002416 = 2^(n^2).
Column 1 is A062738.
T(n,n) = 2^n is the number of binary relations that are a subset of the diagonal relation.
FORMULA
E.g.f. for column k: log(A(x))^k/k! where A(x) is the E.g.f. for A002416
EXAMPLE
2
12 4
432 72 8
61344 3888 288 16
32866560 665280 21600 960 32
MATHEMATICA
a=Sum[2^(n^2) x^n/n!, {n, 0, 10}];
Transpose[Table[Drop[Range[0, 10]! CoefficientList[Series[Log[a]^n/n!, {x, 0, 10}], x], 1], {n, 1, 10}]] // Grid
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, May 28 2011
STATUS
approved