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