A370347 Number T(n,k) of partitions of [3n] into n sets of size 3 having exactly k sets {3j-2,3j-1,3j} (1<=j<=n); triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 9, 0, 1, 252, 27, 0, 1, 14337, 1008, 54, 0, 1, 1327104, 71685, 2520, 90, 0, 1, 182407545, 7962624, 215055, 5040, 135, 0, 1, 34906943196, 1276852815, 27869184, 501795, 8820, 189, 0, 1, 8877242235393, 279255545568, 5107411260, 74317824, 1003590, 14112, 252, 0, 1

Offset

0,4

Links

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

Wikipedia, Partition of a set

Formula

T(n,k) = binomial(n,k) * A370357(n-k).

Sum_{k=1..n} T(n,k) = A370358(n).

T(n,k) mod 9 = A023531(n,k).

Example

T(2,0) = 9: 124|356, 125|346, 126|345, 134|256, 135|246, 136|245, 145|236, 146|235, 156|234.
T(2,2) = 1: 123|456.
Triangle T(n,k) begins:
            1;
            0,          1;
            9,          0,        1;
          252,         27,        0,      1;
        14337,       1008,       54,      0,    1;
      1327104,      71685,     2520,     90,    0,   1;
    182407545,    7962624,   215055,   5040,  135,   0, 1;
  34906943196, 1276852815, 27869184, 501795, 8820, 189, 0, 1;
  ...

Maple

b:= proc(n) option remember; `if`(n<3, [1, 0, 9][n+1],
      9*(n*(n-1)/2*b(n-1)+(n-1)^2*b(n-2)+(n-1)*(n-2)/2*b(n-3)))
    end:
T:= (n, k)-> b(n-k)*binomial(n, k):
seq(seq(T(n, k), k=0..n), n=0..10);

Crossrefs

Row sums give A025035.

Column k=0 gives A370357.

T(n+1,n-1) gives A027468.

T(n+2,n-1) gives 252*A000292.

Cf. A023531, A055140, A370358.

Sequence in context: A189186 A221429 A221507 * A089481 A188593 A065421

Adjacent sequences: A370344 A370345 A370346 * A370348 A370349 A370350

Keyword

nonn,tabl

Author

Alois P. Heinz, Feb 15 2024

Status

approved