A370487 Number of partitions of [3n] into 3 sets of size n having at least one set of consecutive numbers whose maximum (if n>0) is a multiple of n.

1, 1, 7, 28, 103, 376, 1384, 5146, 19303, 72928, 277132, 1058146, 4056232, 15600898, 60174898, 232676278, 901620583, 3500409328, 13612702948, 53017895698, 206769793228, 807386811658, 3156148445578, 12350146091398, 48371405524648, 189615909656626, 743877799422154

Offset: 0

Author

Alois P. Heinz, Feb 19 2024

Keywords

• nonn

Examples

			a(0) = 1: {}|{}|{}.
a(1) = 1: 1|2|3.
a(2) = 7: 12|34|56, 12|35|46, 12|36|45, 13|24|56, 14|23|56, 15|26|34, 16|25|34.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1663
• Wikipedia, Partition of a set

Crossrefs

Row n=3 of A370363.

Cf. A000079, A001700, A003409, A029651, A131577, A255738.

Programs

Maple

a:= n-> `if`(n=0, 1, 3*binomial(2*n-1,n)-2):
seq(a(n), n=0..27);
# second Maple program:
a:= proc(n) option remember; `if`(n<3, 3*n*(n-1)+1, ((15*n^2
      -31*n+12)*a(n-1)-(3*n-2)*(4*n-6)*a(n-2))/((3*n-5)*n))
    end:
seq(a(n), n=0..27);

Formula

a(n) = 3*binomial(2*n-1,n) - 2 for n >= 1, a(0) = 1.

G.f.: ((3*x+1)*(4*x-1)+3*(1-x)*sqrt(1-4*x))/(2*(1-x)*(1-4*x)).

a(n) = A003409(n) - 2 = A029651(n) - 2 = 3*A001700(n-1) - 2 for n >= 1.

a(n) mod 2 = A255738(n+1).

a(n) mod 2 = 1 <=> n in { A131577 }.