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.

1, 0, 1, 0, 1, 3, 0, 1, 18, 12, 0, 1, 97, 198, 55, 0, 1, 530, 2520, 1820, 273, 0, 1, 2973, 29886, 42228, 15300, 1428, 0, 1, 17059, 347907, 859180, 564585, 122094, 7752, 0, 1, 99657, 4048966, 16482191, 17493938, 6577494, 942172, 43263

Offset: 0

Author

Alois P. Heinz, Mar 25 2015

Keywords

• nonn
• tabl

Examples

			T(0,0) = 1: (the empty word).
T(1,1) = 1: aaa.
T(2,1) = 1: aaaaaa.
T(2,2) = 3: aaabbb, aabbba, abbbaa.
T(3,1) = 1: aaaaaaaaa.
T(3,2) = 18: aaaaaabbb, aaaaabbba, aaaabbbaa, aaabaaabb, aaabbaaab, aaabbbaaa, aaabbbbbb, aabaaabba, aabbaaaba, aabbbaaaa, aabbbabbb, aabbbbbba, abaaabbaa, abbaaabaa, abbbaaaaa, abbbaabbb, abbbabbba, abbbbbbaa.
T(3,3) = 12: aaabbbccc, aaabbcccb, aaabcccbb, aabbbaccc, aabbbccca, aabbcccba, aabcccbba, abbbaaccc, abbbaccca, abbbcccaa, abbcccbaa, abcccbbaa.
Triangle T(n,k) begins:
  1;
  0, 1;
  0, 1,     3;
  0, 1,    18,     12;
  0, 1,    97,    198,     55;
  0, 1,   530,   2520,   1820,    273;
  0, 1,  2973,  29886,  42228,  15300,   1428;
  0, 1, 17059, 347907, 859180, 564585, 122094, 7752;
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened

Crossrefs

Columns k=0-10 give: A000007, A057427, A321032, A321033, A321034, A321035, A321036, A321037, A321038, A321039, A321040.

Row sums give A321031.

Main diagonal gives A001764.

T(2n,n) gives A321041.

Cf. A256117, A213028.

Programs

Maple

A:= (n, k)-> `if`(n=0, 1,
    k/n *add(binomial(3*n, j) *(n-j) *(k-1)^j, j=0..n-1)):
T:= (n, k)-> add((-1)^i*A(n, k-i)/(i!*(k-i)!), i=0..k):
seq(seq(T(n, k), k=0..n), n=0..10);

Formula

T(n,k) = Sum_{i=0..k} (-1)^i * A213028(n,k-i) / (i!*(k-i)!).