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A175757
Triangular array read by rows: T(n,k) is the number of blocks of size k in all set partitions of {1,2,...,n}.
4
1, 2, 1, 6, 3, 1, 20, 12, 4, 1, 75, 50, 20, 5, 1, 312, 225, 100, 30, 6, 1, 1421, 1092, 525, 175, 42, 7, 1, 7016, 5684, 2912, 1050, 280, 56, 8, 1, 37260, 31572, 17052, 6552, 1890, 420, 72, 9, 1, 211470, 186300, 105240, 42630, 13104, 3150, 600, 90, 10, 1
OFFSET
1,2
COMMENTS
The row sums of this triangle equal A005493. Equals A056857 without its leftmost column.
T(n,k) = binomial(n,k)*B(n-k) where B is the Bell number.
LINKS
FORMULA
E.g.f. for column k is x^k/k!*exp(exp(x)-1).
Sum_{k=1..n} k * T(n,k) = A070071(n). - Alois P. Heinz, Mar 03 2020
EXAMPLE
The set {1,2,3} has 5 partitions, {{1, 2, 3}}, {{2, 3}, {1}}, {{1, 3}, {2}}, {{1, 2}, {3}}, and {{2}, {3}, {1}}, and there are a total of 3 blocks of size 2, so T(3,2)=3.
Triangle begins:
1;
2, 1;
6, 3, 1;
20, 12, 4, 1;
75, 50, 20, 5, 1;
312, 225, 100, 30, 6, 1;
...
MAPLE
b:= proc(n) option remember; `if`(n=0, [1, 0],
add((p-> p+[0, p[1]*x^j])(b(n-j)*
binomial(n-1, j-1)), j=1..n))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n)[2]):
seq(T(n), n=1..12); # Alois P. Heinz, Apr 24 2017
MATHEMATICA
Table[Table[Length[Select[Level[SetPartitions[m], {2}], Length[#]==n&]], {n, 1, m}], {m, 1, 10}]//Grid
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Dec 04 2010
STATUS
approved