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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}. 3


%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).

%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;

%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, A105479, A105480, A105481, A105482, A285595.

%K nonn,tabl

%O 1,2

%A _Geoffrey Critzer_, Dec 04 2010

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Last modified December 13 22:55 EST 2019. Contains 329974 sequences. (Running on oeis4.)