A370358 Number of partitions of [3n] into n sets of size 3 having at least one set {3j-2,3j-1,3j} (1<=j<=n).

0, 1, 1, 28, 1063, 74296, 8182855, 1305232804, 284438292607, 81167321350432, 29367491879327959, 13135455977606994340, 7116140280642196449151, 4591529352468711908776288, 3479040085783649820897765223, 3058744793640846605215609362436

Offset: 0

Author

Alois P. Heinz, Feb 16 2024

Keywords

• nonn

Examples

			a(1) = 1: 123.
a(2) = 1: 123|456.
a(3) = 28: 123|456|789, 123|457|689, 123|458|679, 123|459|678, 123|467|589, 123|468|579, 123|469|578, 123|478|569, 123|479|568, 123|489|567, 124|356|789, 125|346|789, 126|345|789, 127|389|456, 128|379|456, 129|378|456, 134|256|789, 135|246|789, 136|245|789, 137|289|456, 138|279|456, 139|278|456, 145|236|789, 146|235|789, 156|234|789, 178|239|456, 179|238|456, 189|237|456.
		

Links

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

Crossrefs

Cf. A025035, A057427, A059841, A370347, A370357.

Column k=3 of A370363.

Programs

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:
a:= n-> (3*n)!/(n!*(3!)^n)-b(n):
seq(a(n), n=0..20);

Formula

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

a(n) = A025035(n) - A370357(n).

a(n) = Sum_{k=1..n} A370347(n,k).

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

a(n) mod 9 = A057427(n).