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

0,6

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:
  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;

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);

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.

Sequence in context: A304336 A287315 A350212 * A022695 A278325 A364527

Adjacent sequences: A256308 A256309 A256310 * A256312 A256313 A256314

Keyword

nonn,tabl

Author

Alois P. Heinz, Mar 25 2015

Status

approved