A370357 Number of partitions of [3n] into n sets of size 3 avoiding any set {3j-2,3j-1,3j} (1<=j<=n).

1, 0, 9, 252, 14337, 1327104, 182407545, 34906943196, 8877242235393, 2896378850249568, 1179516253790272041, 586470881874514605660, 349649630741370155550849, 246214807676005971547223712, 202182156277565590613022234777, 191496746966087534845272710637564

Offset: 0

Author

Alois P. Heinz, Feb 16 2024

Keywords

• nonn

Examples

			a(0) = 1: the empty partition satisfies the condition.
a(1) = 0: 123 is not counted.
a(2) = 9: 124|356, 125|346, 126|345, 134|256, 135|246, 136|245, 145|236, 146|235, 156|234 are counted. 123|456 is not counted.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..223
• Wikipedia, Partition of a set

Crossrefs

Column k=0 of A370347.

Column k=3 of A370366.

Cf. A000007, A025035, A059841, A370358.

Programs

Maple

a:= proc(n) option remember; `if`(n<3, [1, 0, 9][n+1],
      9*(n*(n-1)/2*a(n-1)+(n-1)^2*a(n-2)+(n-1)*(n-2)/2*a(n-3)))
    end:
seq(a(n), n=0..20);

Formula

a(n) = Sum_{j=0..n} (-1)^(n-j) * binomial(n,j) * A025035(j).

a(n) = A025035(n) - A370358(n).

a(n) mod 9 = A000007(n).

a(n) mod 2 = A059841(n).