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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.

%I A370487 #20 Feb 19 2024 18:26:32
%S A370487 1,1,7,28,103,376,1384,5146,19303,72928,277132,1058146,4056232,
%T A370487 15600898,60174898,232676278,901620583,3500409328,13612702948,
%U A370487 53017895698,206769793228,807386811658,3156148445578,12350146091398,48371405524648,189615909656626,743877799422154
%N 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.
%H A370487 Alois P. Heinz, <a href="/A370487/b370487.txt">Table of n, a(n) for n = 0..1663</a>
%H A370487 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A370487 a(n) = 3*binomial(2*n-1,n) - 2 for n >= 1, a(0) = 1.
%F A370487 G.f.: ((3*x+1)*(4*x-1)+3*(1-x)*sqrt(1-4*x))/(2*(1-x)*(1-4*x)).
%F A370487 a(n) = A003409(n) - 2 = A029651(n) - 2 = 3*A001700(n-1) - 2 for n >= 1.
%F A370487 a(n) mod 2 = A255738(n+1).
%F A370487 a(n) mod 2 = 1 <=> n in { A131577 }.
%e A370487 a(0) = 1: {}|{}|{}.
%e A370487 a(1) = 1: 1|2|3.
%e A370487 a(2) = 7: 12|34|56, 12|35|46, 12|36|45, 13|24|56, 14|23|56, 15|26|34, 16|25|34.
%p A370487 a:= n-> `if`(n=0, 1, 3*binomial(2*n-1,n)-2):
%p A370487 seq(a(n), n=0..27);
%p A370487 # second Maple program:
%p A370487 a:= proc(n) option remember; `if`(n<3, 3*n*(n-1)+1, ((15*n^2
%p A370487       -31*n+12)*a(n-1)-(3*n-2)*(4*n-6)*a(n-2))/((3*n-5)*n))
%p A370487     end:
%p A370487 seq(a(n), n=0..27);
%Y A370487 Row n=3 of A370363.
%Y A370487 Cf. A000079, A001700, A003409, A029651, A131577, A255738.
%K A370487 nonn
%O A370487 0,3
%A A370487 _Alois P. Heinz_, Feb 19 2024