A370357
Number of partitions of [3n] into n sets of size 3 avoiding any set {3j-2,3j-1,3j} (1<=j<=n).
Original entry on oeis.org
1, 0, 9, 252, 14337, 1327104, 182407545, 34906943196, 8877242235393, 2896378850249568, 1179516253790272041, 586470881874514605660, 349649630741370155550849, 246214807676005971547223712, 202182156277565590613022234777, 191496746966087534845272710637564
Offset: 0
a(0) = 1: the empty partition satisfies the condition.
a(1) = 0: 123 is not counted.
a(2) = 9: 124|356, 125|346, 126|345, 134|256, 135|246, 136|245, 145|236, 146|235, 156|234 are counted. 123|456 is not counted.
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a:= proc(n) option remember; `if`(n<3, [1, 0, 9][n+1],
9*(n*(n-1)/2*a(n-1)+(n-1)^2*a(n-2)+(n-1)*(n-2)/2*a(n-3)))
end:
seq(a(n), n=0..20);
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).
Original entry on oeis.org
0, 1, 1, 28, 1063, 74296, 8182855, 1305232804, 284438292607, 81167321350432, 29367491879327959, 13135455977606994340, 7116140280642196449151, 4591529352468711908776288, 3479040085783649820897765223, 3058744793640846605215609362436
Offset: 0
a(1) = 1: 123.
a(2) = 1: 123|456.
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.
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b:= proc(n) option remember; `if`(n<3, [1, 0, 9][n+1],
9*(n*(n-1)/2*b(n-1)+(n-1)^2*b(n-2)+(n-1)*(n-2)/2*b(n-3)))
end:
a:= n-> (3*n)!/(n!*(3!)^n)-b(n):
seq(a(n), n=0..20);
Showing 1-2 of 2 results.