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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.
2
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
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
KEYWORD
nonn
AUTHOR
Alois P. Heinz, Feb 19 2024
STATUS
approved