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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. 3
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 (list; table; graph; refs; listen; history; text; internal format)
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

Main diagonal gives A001764.

Cf. A256117, A213028.

Sequence in context: A279031 A304336 A287315 * A022695 A278325 A226780

Adjacent sequences:  A256308 A256309 A256310 * A256312 A256313 A256314

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Mar 25 2015

STATUS

approved

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Last modified October 16 02:45 EDT 2018. Contains 316252 sequences. (Running on oeis4.)