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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.

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

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OFFSET

0,6

LINKS

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

FORMULA

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

EXAMPLE

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:

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;

MAPLE

A:= (n, k)-> `if`(n=0, 1,

k/n *add(binomial(3*n, j) *(n-j) *(k-1)^j, j=0..n-1)):