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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}.

%I A175757 #32 Mar 03 2020 16:04:21
%S A175757 1,2,1,6,3,1,20,12,4,1,75,50,20,5,1,312,225,100,30,6,1,1421,1092,525,
%T A175757 175,42,7,1,7016,5684,2912,1050,280,56,8,1,37260,31572,17052,6552,
%U A175757 1890,420,72,9,1,211470,186300,105240,42630,13104,3150,600,90,10,1
%N 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}.
%C A175757 The row sums of this triangle equal A005493. Equals A056857 without its leftmost column.
%C A175757 T(n,k) = binomial(n,k)*B(n-k) where B is the Bell number.
%H A175757 Alois P. Heinz, <a href="/A175757/b175757.txt">Rows n = 1..141, flattened</a>
%F A175757 E.g.f. for column k is x^k/k!*exp(exp(x)-1).
%F A175757 Sum_{k=1..n} k * T(n,k) = A070071(n). - _Alois P. Heinz_, Mar 03 2020
%e A175757 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.
%e A175757 Triangle begins:
%e A175757     1;
%e A175757     2,   1;
%e A175757     6,   3,   1;
%e A175757    20,  12,   4,  1;
%e A175757    75,  50,  20,  5, 1;
%e A175757   312, 225, 100, 30, 6, 1;
%e A175757   ...
%p A175757 b:= proc(n) option remember; `if`(n=0, [1, 0],
%p A175757       add((p-> p+[0, p[1]*x^j])(b(n-j)*
%p A175757       binomial(n-1, j-1)), j=1..n))
%p A175757     end:
%p A175757 T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n)[2]):
%p A175757 seq(T(n), n=1..12);  # _Alois P. Heinz_, Apr 24 2017
%t A175757 Table[Table[Length[Select[Level[SetPartitions[m],{2}],Length[#]==n&]],{n,1,m}],{m,1,10}]//Grid
%Y A175757 Cf. A000110, A056860, A052889, A070071, A105479, A105480, A105481, A105482, A285595.
%K A175757 nonn,tabl
%O A175757 1,2
%A A175757 _Geoffrey Critzer_, Dec 04 2010