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A351638 Number of length n word structures with all distinct run-lengths using an infinite alphabet. 6
1, 1, 1, 3, 3, 5, 17, 19, 31, 45, 177, 191, 335, 469, 733, 2679, 3063, 5129, 7445, 11431, 15667, 59025, 65301, 112379, 159827, 248185, 336913, 505683, 1660611, 1909901, 3184601, 4576771, 6994351, 9606093, 14229033, 19085255, 61388207, 69587029, 116257501, 164298495, 252820047 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,4
COMMENTS
Permuting the symbols will not change the structure.
Equivalently, a(n) 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 and every run has a different length.
LINKS
EXAMPLE
The a(3) = 3 words are 111, 112, 122.
The a(4) = 3 words are 1111, 1112, 1222. The word 1122 is not included because both runs have the same length.
The a(6) = 17 words are 111111, 111112, 111122, 111211, 111221, 112111, 112221, 112222, 122111, 122211, 122222, 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]}
seq(n)={my(u=P(n)); concat([1], sum(k=1, n, R(u, k)*sum(r=k, n, binomial(r, k)*(-1)^(r-k)/r!) ))}
CROSSREFS
Row sums of A351637.
Sequence in context: A370973 A139431 A351639 * A369345 A144423 A128636
KEYWORD
nonn
AUTHOR
Andrew Howroyd, Feb 15 2022
STATUS
approved

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Last modified April 15 14:37 EDT 2024. Contains 371689 sequences. (Running on oeis4.)