A370363
Number A(n,k) of partitions of [k*n] into n sets of size k having at least one 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
0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 7, 1, 1, 0, 1, 1, 28, 45, 1, 1, 0, 1, 1, 103, 1063, 401, 1, 1, 0, 1, 1, 376, 22893, 74296, 4355, 1, 1, 0, 1, 1, 1384, 503751, 13080721, 8182855, 56127, 1, 1, 0, 1, 1, 5146, 11432655, 2443061876, 15237712355, 1305232804, 836353, 1, 1
Offset: 0
A(3,2) = 7: 12|34|56, 12|35|46, 12|36|45, 13|24|56, 14|23|56, 15|26|34, 16|25|34.
Square array A(n,k) begins:
0, 0, 0, 0, 0, 0, ...
1, 1, 1, 1, 1, 1, ...
1, 1, 1, 1, 1, 1, ...
1, 1, 7, 28, 103, 376, ...
1, 1, 45, 1063, 22893, 503751, ...
1, 1, 401, 74296, 13080721, 2443061876, ...
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A:= proc(n, k) option remember; `if`(k=0, signum(n), add(
(-1)^(n-j+1)*binomial(n, j)*(k*j)!/(j!*k!^j), j=0..n-1))
end:
seq(seq(A(n, d-n), n=0..d), d=0..10);
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);
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.