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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.

%I A370347 #19 Feb 17 2024 14:44:58
%S A370347 1,0,1,9,0,1,252,27,0,1,14337,1008,54,0,1,1327104,71685,2520,90,0,1,
%T A370347 182407545,7962624,215055,5040,135,0,1,34906943196,1276852815,
%U A370347 27869184,501795,8820,189,0,1,8877242235393,279255545568,5107411260,74317824,1003590,14112,252,0,1
%N 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.
%H A370347 Alois P. Heinz, <a href="/A370347/b370347.txt">Rows n = 0..140, flattened</a>
%H A370347 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A370347 T(n,k) = binomial(n,k) * A370357(n-k).
%F A370347 Sum_{k=1..n} T(n,k) = A370358(n).
%F A370347 T(n,k) mod 9 = A023531(n,k).
%e A370347 T(2,0) = 9: 124|356, 125|346, 126|345, 134|256, 135|246, 136|245, 145|236, 146|235, 156|234.
%e A370347 T(2,2) = 1: 123|456.
%e A370347 Triangle T(n,k) begins:
%e A370347             1;
%e A370347             0,          1;
%e A370347             9,          0,        1;
%e A370347           252,         27,        0,      1;
%e A370347         14337,       1008,       54,      0,    1;
%e A370347       1327104,      71685,     2520,     90,    0,   1;
%e A370347     182407545,    7962624,   215055,   5040,  135,   0, 1;
%e A370347   34906943196, 1276852815, 27869184, 501795, 8820, 189, 0, 1;
%e A370347   ...
%p A370347 b:= proc(n) option remember; `if`(n<3, [1, 0, 9][n+1],
%p A370347       9*(n*(n-1)/2*b(n-1)+(n-1)^2*b(n-2)+(n-1)*(n-2)/2*b(n-3)))
%p A370347     end:
%p A370347 T:= (n, k)-> b(n-k)*binomial(n, k):
%p A370347 seq(seq(T(n, k), k=0..n), n=0..10);
%Y A370347 Row sums give A025035.
%Y A370347 Column k=0 gives A370357.
%Y A370347 T(n+1,n-1) gives A027468.
%Y A370347 T(n+2,n-1) gives 252*A000292.
%Y A370347 Cf. A023531, A055140, A370358.
%K A370347 nonn,tabl
%O A370347 0,4
%A A370347 _Alois P. Heinz_, Feb 15 2024