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

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1663

Wikipedia, Partition of a set

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 }.

Example

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.

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);

Crossrefs

Row n=3 of A370363.

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

Sequence in context: A241400 A219411 A224404 * A331197 A024207 A000416

Adjacent sequences: A370483 A370485 A370486 * A370488 A370489 A370490

Keyword

nonn

Author

Alois P. Heinz, Feb 19 2024

Status

approved