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A351641 Triangle read by rows: T(n,k) is the number of length n word structures with all distinct runs using exactly k different symbols. 6
1, 0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 5, 3, 1, 0, 1, 8, 12, 4, 1, 0, 1, 17, 28, 22, 5, 1, 0, 1, 26, 81, 68, 35, 6, 1, 0, 1, 45, 177, 251, 135, 51, 7, 1, 0, 1, 76, 410, 704, 610, 236, 70, 8, 1, 0, 1, 121, 906, 2068, 2086, 1266, 378, 92, 9, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,9
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 all runs are distinct.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1325 (rows 0..50)
FORMULA
T(n,k) = A351640(n,k)/k!.
EXAMPLE
Triangle begins:
1;
0, 1;
0, 1, 1;
0, 1, 2, 1;
0, 1, 5, 3, 1;
0, 1, 8, 12, 4, 1;
0, 1, 17, 28, 22, 5, 1;
0, 1, 26, 81, 68, 35, 6, 1;
0, 1, 45, 177, 251, 135, 51, 7, 1;
...
The T(4,1) = 1 word is 1111.
The T(4,2) = 5 words are 1112, 1121, 1122, 1211, 1222.
The T(4,3) = 3 words are 1123, 1223, 1233.
The T(4,4) = 1 word is 1234.
PROG
(PARI) \\ here LahI is A111596 as row polynomials.
LahI(n, y)={sum(k=1, n, y^k*(-1)^(n-k)*(n!/k!)*binomial(n-1, k-1))}
S(n)={my(p=prod(k=1, n, 1 + y*x^k + O(x*x^n))); 1 + sum(i=1, (sqrtint(8*n+1)-1)\2, polcoef(p, i, y)*LahI(i, y))}
R(q)={[subst(serlaplace(p), y, 1) | p<-Vec(q)]}
T(n)={my(q=S(n), v=concat([1], sum(k=1, n, R(q^k-1)*sum(r=k, n, y^r*binomial(r, k)*(-1)^(r-k)/r!) ))); [Vecrev(p) | p<-v]}
{ my(A=T(10)); for(n=1, #A, print(A[n])) }
CROSSREFS
Row sums are A351642.
Partial row sums include A000007, A000012, A351018, A351644.
Column k=3 is A351643.
Sequence in context: A263339 A244372 A119331 * A291883 A361957 A239145
KEYWORD
nonn,tabl
AUTHOR
Andrew Howroyd, Feb 15 2022
STATUS
approved

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Last modified February 20 21:52 EST 2024. Contains 370219 sequences. (Running on oeis4.)