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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. 15
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
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
Row sums give A321031.
Main diagonal gives A001764.
T(2n,n) gives A321041.
Sequence in context: A304336 A287315 A350212 * A022695 A278325 A364527
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Mar 25 2015
STATUS
approved

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Last modified May 17 23:39 EDT 2024. Contains 372608 sequences. (Running on oeis4.)