A175757 - OEIS

%I #32 Mar 03 2020 16:04:21

%S 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 175,42,7,1,7016,5684,2912,1050,280,56,8,1,37260,31572,17052,6552,

%U 1890,420,72,9,1,211470,186300,105240,42630,13104,3150,600,90,10,1

%N 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 The row sums of this triangle equal A005493. Equals A056857 without its leftmost column.

%C T(n,k) = binomial(n,k)*B(n-k) where B is the Bell number.

%H Alois P. Heinz, <a href="/A175757/b175757.txt">Rows n = 1..141, flattened</a>

%F E.g.f. for column k is x^k/k!*exp(exp(x)-1).

%F Sum_{k=1..n} k * T(n,k) = A070071(n). - _Alois P. Heinz_, Mar 03 2020

%e 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 Triangle begins:

%e     1;

%e     2,   1;

%e     6,   3,   1;

%e    20,  12,   4,  1;

%e    75,  50,  20,  5, 1;

%e   312, 225, 100, 30, 6, 1;

%e   ...

%p b:= proc(n) option remember; `if`(n=0, [1, 0],

%p       add((p-> p+[0, p[1]*x^j])(b(n-j)*

%p       binomial(n-1, j-1)), j=1..n))

%p     end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b(n)[2]):

%p seq(T(n), n=1..12);  # _Alois P. Heinz_, Apr 24 2017

%t Table[Table[Length[Select[Level[SetPartitions[m],{2}],Length[#]==n&]],{n,1,m}],{m,1,10}]//Grid

%Y Cf. A000110, A056860, A052889, A070071, A105479, A105480, A105481, A105482, A285595.

%K nonn,tabl

%O 1,2

%A _Geoffrey Critzer_, Dec 04 2010