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A327884 Number T(n,k) of set partitions of [n] such that at least one of the block sizes is k or k=0; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 6
1, 1, 1, 2, 1, 1, 5, 4, 3, 1, 15, 11, 9, 4, 1, 52, 41, 35, 20, 5, 1, 203, 162, 150, 90, 30, 6, 1, 877, 715, 672, 455, 175, 42, 7, 1, 4140, 3425, 3269, 2352, 1015, 280, 56, 8, 1, 21147, 17722, 17271, 13132, 6237, 1890, 420, 72, 9, 1, 115975, 98253, 97155, 76540, 39480, 12978, 3150, 600, 90, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

LINKS

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

Wikipedia, Iverson bracket

Wikipedia, Partition of a set

FORMULA

E.g.f. of column k: exp(exp(x)-1) - [k>0] * exp(exp(x)-1-x^k/k!).

T(n,0) - T(n,1) = A000296(n).

EXAMPLE

T(4,1) = 11: 123|4, 124|3, 12|3|4, 134|2, 13|2|4, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.

T(4,2) = 9: 12|34, 12|3|4, 13|24, 13|2|4, 14|23, 1|23|4, 14|2|3, 1|24|3, 1|2|34.

T(4,3) = 4: 123|4, 124|3, 134|2, 1|234.

T(4,4) = 1: 1234.

T(5,1) = 41: 1234|5, 1235|4, 123|4|5, 1245|3, 124|3|5, 12|34|5, 125|3|4, 12|35|4, 12|3|45, 12|3|4|5, 1345|2, 134|2|5, 13|24|5, 135|2|4, 13|25|4, 13|2|45, 13|2|4|5, 14|23|5, 1|2345, 1|234|5, 15|23|4, 1|235|4, 1|23|45, 1|23|4|5, 145|2|3, 14|25|3, 14|2|35, 14|2|3|5, 15|24|3, 1|245|3, 1|24|35, 1|24|3|5, 15|2|34, 1|25|34, 1|2|345, 1|2|34|5, 15|2|3|4, 1|25|3|4, 1|2|35|4, 1|2|3|45, 1|2|3|4|5.

Triangle T(n,k) begins:

      1;

      1,     1;

      2,     1,     1;

      5,     4,     3,     1;

     15,    11,     9,     4,    1;

     52,    41,    35,    20,    5,    1;

    203,   162,   150,    90,   30,    6,   1;

    877,   715,   672,   455,  175,   42,   7,  1;

   4140,  3425,  3269,  2352, 1015,  280,  56,  8, 1;

  21147, 17722, 17271, 13132, 6237, 1890, 420, 72, 9, 1;

  ...

MAPLE

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

      `if`(j=k, 0, b(n-j, k)*binomial(n-1, j-1)), j=1..n))

    end:

T:= (n, k)-> b(n, 0)-`if`(k=0, 0, b(n, k)):

seq(seq(T(n, k), k=0..n), n=0..11);

MATHEMATICA

b[n_, k_] := b[n, k] = If[n == 0, 1, Sum[If[j == k, 0, b[n - j, k] Binomial[ n - 1, j - 1]], {j, 1, n}]];

T[n_, k_] := b[n, 0] - If[k == 0, 0, b[n, k]];

Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-Fran├žois Alcover, Apr 30 2020, after Alois P. Heinz *)

CROSSREFS

Columns k=0-3 give: A000110, A000296(n+1), A327885, A328153.

T(2n,n) gives A276961.

Cf. A080510, A327869.

Sequence in context: A220738 A284732 A327483 * A050145 A222573 A222679

Adjacent sequences:  A327881 A327882 A327883 * A327885 A327886 A327887

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Sep 28 2019

STATUS

approved

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Last modified October 23 00:39 EDT 2021. Contains 348211 sequences. (Running on oeis4.)