login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A323099 Number T(n,k) of colored set partitions of [n] where exactly k colors are used for the elements; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 3
1, 0, 1, 0, 2, 4, 0, 5, 30, 30, 0, 15, 210, 540, 360, 0, 52, 1560, 7800, 12480, 6240, 0, 203, 12586, 109620, 316680, 365400, 146160, 0, 877, 110502, 1583862, 7366800, 14733600, 13260240, 4420080, 0, 4140, 1051560, 23995440, 169011360, 521640000, 792892800, 584236800, 166924800 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

LINKS

Alois P. Heinz, Rows n = 0..140, flattened

Wikipedia, Partition of a set

FORMULA

T(n,k) = Bell(n) * Sum_{i=0..k} (k-i)^n * (-1)^i * C(k,i).

T(n,k) = Bell(n) * A131689(n,k).

T(n,k) = Bell(n) * Stirling2(n,k) * k!.

EXAMPLE

Triangle T(n,k) begins:

  1;

  0,   1;

  0,   2,      4;

  0,   5,     30,      30;

  0,  15,    210,     540,     360;

  0,  52,   1560,    7800,   12480,     6240;

  0, 203,  12586,  109620,  316680,   365400,   146160;

  0, 877, 110502, 1583862, 7366800, 14733600, 13260240, 4420080;

  ...

MAPLE

A:= proc(n, k) option remember; `if`(n=0, 1, add(

      A(n-j, k)*binomial(n-1, j-1)*k^j, j=1..n))

    end:

T:= (n, k)-> add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k):

seq(seq(T(n, k), k=0..n), n=0..10);

# second Maple program:

T:= (n, k)-> combinat[bell](n)*Stirling2(n, k)*k!:

seq(seq(T(n, k), k=0..n), n=0..10);

CROSSREFS

Columns k=0-1 give: A000007, A000110 (for n>0).

Row sums give A121017.

Main diagonal gives A137341.

Cf. A000110, A000142, A008277, A048993, A019538, A131689.

Sequence in context: A274086 A255982 A256061 * A002652 A202541 A070676

Adjacent sequences:  A323096 A323097 A323098 * A323100 A323101 A323102

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Aug 30 2019

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified July 14 16:21 EDT 2020. Contains 335729 sequences. (Running on oeis4.)