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A256311 Number T(n,k) of length 3n words such that all letters of the k-ary alphabet occur at least once and are introduced in ascending order and which can be built by repeatedly inserting triples of identical letters into the initially empty word; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A256311 #32 Oct 26 2018 14:59:18
%S A256311 1,0,1,0,1,3,0,1,18,12,0,1,97,198,55,0,1,530,2520,1820,273,0,1,2973,
%T A256311 29886,42228,15300,1428,0,1,17059,347907,859180,564585,122094,7752,0,
%U A256311 1,99657,4048966,16482191,17493938,6577494,942172,43263
%N A256311 Number T(n,k) of length 3n words such that all letters of the k-ary alphabet occur at least once and are introduced in ascending order and which can be built by repeatedly inserting triples of identical letters into the initially empty word; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H A256311 Alois P. Heinz, <a href="/A256311/b256311.txt">Rows n = 0..140, flattened</a>
%F A256311 T(n,k) = Sum_{i=0..k} (-1)^i * A213028(n,k-i) / (i!*(k-i)!).
%e A256311 T(0,0) = 1: (the empty word).
%e A256311 T(1,1) = 1: aaa.
%e A256311 T(2,1) = 1: aaaaaa.
%e A256311 T(2,2) = 3: aaabbb, aabbba, abbbaa.
%e A256311 T(3,1) = 1: aaaaaaaaa.
%e A256311 T(3,2) = 18: aaaaaabbb, aaaaabbba, aaaabbbaa, aaabaaabb, aaabbaaab, aaabbbaaa, aaabbbbbb, aabaaabba, aabbaaaba, aabbbaaaa, aabbbabbb, aabbbbbba, abaaabbaa, abbaaabaa, abbbaaaaa, abbbaabbb, abbbabbba, abbbbbbaa.
%e A256311 T(3,3) = 12: aaabbbccc, aaabbcccb, aaabcccbb, aabbbaccc, aabbbccca, aabbcccba, aabcccbba, abbbaaccc, abbbaccca, abbbcccaa, abbcccbaa, abcccbbaa.
%e A256311 Triangle T(n,k) begins:
%e A256311   1;
%e A256311   0, 1;
%e A256311   0, 1,     3;
%e A256311   0, 1,    18,     12;
%e A256311   0, 1,    97,    198,     55;
%e A256311   0, 1,   530,   2520,   1820,    273;
%e A256311   0, 1,  2973,  29886,  42228,  15300,   1428;
%e A256311   0, 1, 17059, 347907, 859180, 564585, 122094, 7752;
%p A256311 A:= (n, k)-> `if`(n=0, 1,
%p A256311     k/n *add(binomial(3*n, j) *(n-j) *(k-1)^j, j=0..n-1)):
%p A256311 T:= (n, k)-> add((-1)^i*A(n, k-i)/(i!*(k-i)!), i=0..k):
%p A256311 seq(seq(T(n, k), k=0..n), n=0..10);
%Y A256311 Columns k=0-10 give: A000007, A057427, A321032, A321033, A321034, A321035, A321036, A321037, A321038, A321039, A321040.
%Y A256311 Row sums give A321031.
%Y A256311 Main diagonal gives A001764.
%Y A256311 T(2n,n) gives A321041.
%Y A256311 Cf. A256117, A213028.
%K A256311 nonn,tabl
%O A256311 0,6
%A A256311 _Alois P. Heinz_, Mar 25 2015