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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
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 (list; table; graph; refs; listen; history; text; internal format)
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

Alois P. Heinz, Rows n = 1..141, flattened

FORMULA

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

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

CROSSREFS

Cf. A000110, A056860, A052889, A105479, A105480, A105481, A105482, A285595.

Sequence in context: A246971 A092392 A128741 * A060539 A163269 A103905

Adjacent sequences:  A175754 A175755 A175756 * A175758 A175759 A175760

KEYWORD

nonn,tabl

AUTHOR

Geoffrey Critzer, Dec 04 2010

STATUS

approved

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Last modified November 22 10:59 EST 2019. Contains 329389 sequences. (Running on oeis4.)