A370366
Number A(n,k) of partitions of [k*n] into n sets of size k having no set of consecutive numbers whose maximum (if k>0) is a multiple of k; square array A(n,k), n>=0, k>=0, read by antidiagonals.
Original entry on oeis.org
1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 1, 0, 9, 8, 0, 0, 1, 0, 34, 252, 60, 0, 0, 1, 0, 125, 5672, 14337, 544, 0, 0, 1, 0, 461, 125750, 2604732, 1327104, 6040, 0, 0, 1, 0, 1715, 2857472, 488360625, 2533087904, 182407545, 79008, 0, 0
Offset: 0
A(2,3) = 9: 124|356, 125|346, 126|345, 134|256, 135|246, 136|245, 145|236, 146|235, 156|234.
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, ...
0, 0, 0, 0, 0, 0, ...
0, 0, 2, 9, 34, 125, ...
0, 0, 8, 252, 5672, 125750, ...
0, 0, 60, 14337, 2604732, 488360625, ...
0, 0, 544, 1327104, 2533087904, 5192229797500, ...
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A:= proc(n, k) `if`(k=0,`if`(n=0, 1, 0), add(
(-1)^(n-j)*binomial(n, j)*(k*j)!/(j!*k!^j), j=0..n))
end:
seq(seq(A(n, d-n), n=0..d), d=0..10);
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);
A370347
Number T(n,k) of partitions of [3n] into n sets of size 3 having exactly k sets {3j-2,3j-1,3j} (1<=j<=n); triangle T(n,k), n>=0, 0<=k<=n, read by rows.
Original entry on oeis.org
1, 0, 1, 9, 0, 1, 252, 27, 0, 1, 14337, 1008, 54, 0, 1, 1327104, 71685, 2520, 90, 0, 1, 182407545, 7962624, 215055, 5040, 135, 0, 1, 34906943196, 1276852815, 27869184, 501795, 8820, 189, 0, 1, 8877242235393, 279255545568, 5107411260, 74317824, 1003590, 14112, 252, 0, 1
Offset: 0
T(2,0) = 9: 124|356, 125|346, 126|345, 134|256, 135|246, 136|245, 145|236, 146|235, 156|234.
T(2,2) = 1: 123|456.
Triangle T(n,k) begins:
1;
0, 1;
9, 0, 1;
252, 27, 0, 1;
14337, 1008, 54, 0, 1;
1327104, 71685, 2520, 90, 0, 1;
182407545, 7962624, 215055, 5040, 135, 0, 1;
34906943196, 1276852815, 27869184, 501795, 8820, 189, 0, 1;
...
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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:
T:= (n, k)-> b(n-k)*binomial(n, k):
seq(seq(T(n, k), k=0..n), n=0..10);
Showing 1-3 of 3 results.