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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).

%I A370358 #17 Feb 17 2024 14:33:33
%S A370358 0,1,1,28,1063,74296,8182855,1305232804,284438292607,81167321350432,
%T A370358 29367491879327959,13135455977606994340,7116140280642196449151,
%U A370358 4591529352468711908776288,3479040085783649820897765223,3058744793640846605215609362436
%N 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).
%H A370358 Alois P. Heinz, <a href="/A370358/b370358.txt">Table of n, a(n) for n = 0..223</a>
%H A370358 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A370358 a(n) = Sum_{j=0..n-1} (-1)^(n-j+1) * binomial(n,j) * A025035(j).
%F A370358 a(n) = A025035(n) - A370357(n).
%F A370358 a(n) = Sum_{k=1..n} A370347(n,k).
%F A370358 a(n) mod 2 = A059841(n) for n>=2.
%F A370358 a(n) mod 9 = A057427(n).
%e A370358 a(1) = 1: 123.
%e A370358 a(2) = 1: 123|456.
%e A370358 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.
%p A370358 b:= proc(n) option remember; `if`(n<3, [1, 0, 9][n+1],
%p A370358       9*(n*(n-1)/2*b(n-1)+(n-1)^2*b(n-2)+(n-1)*(n-2)/2*b(n-3)))
%p A370358     end:
%p A370358 a:= n-> (3*n)!/(n!*(3!)^n)-b(n):
%p A370358 seq(a(n), n=0..20);
%Y A370358 Cf. A025035, A057427, A059841, A370347, A370357.
%Y A370358 Column k=3 of A370363.
%K A370358 nonn
%O A370358 0,4
%A A370358 _Alois P. Heinz_, Feb 16 2024