OFFSET
0,8
COMMENTS
Permuting the symbols will not change the structure.
Equivalently, T(n,k) is the number of restricted growth strings [s(0), s(1), ..., s(n-1)] where s(0)=0 and s(i) <= 1 + max(prefix) for i >= 1, the maximum value is k and every run has a different length.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..958 (rows 0..100)
FORMULA
EXAMPLE
Triangle begins:
1;
0, 1;
0, 1;
0, 1, 2;
0, 1, 2;
0, 1, 4;
0, 1, 10, 6;
0, 1, 12, 6;
0, 1, 18, 12;
0, 1, 26, 18;
0, 1, 56, 96, 24;
0, 1, 64, 102, 24;
0, 1, 100, 186, 48;
0, 1, 132, 264, 72;
...
The T(6,1) = 1 word is 111111.
The T(6,2) = 10 words are 111112, 111122, 111211, 111221, 112111, 112221, 112222, 122111, 122211, 122222.
The T(6,3) = 6 words are 111223, 111233, 112333, 112223, 122333, 122233.
PROG
(PARI)
P(n) = {Vec(-1 + prod(k=1, n, 1 + y*x^k + O(x*x^n)))}
R(u, k) = {k*[subst(serlaplace(p)/y, y, k-1) | p<-u]}
T(n)={my(u=P(n), v=concat([1], sum(k=1, n, R(u, k)*sum(r=k, n, y^r*binomial(r, k)*(-1)^(r-k)/r!) ))); [Vecrev(p) | p<-v]}
{ my(A=T(16)); for(n=1, #A, print(A[n])) }
KEYWORD
nonn,tabf,changed
AUTHOR
Andrew Howroyd, Feb 15 2022
STATUS
approved