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A351638 Number of length n word structures with all distinct run-lengths using an infinite alphabet.

%I A351638 #13 Jan 29 2023 13:31:41
%S A351638 1,1,1,3,3,5,17,19,31,45,177,191,335,469,733,2679,3063,5129,7445,
%T A351638 11431,15667,59025,65301,112379,159827,248185,336913,505683,1660611,
%U A351638 1909901,3184601,4576771,6994351,9606093,14229033,19085255,61388207,69587029,116257501,164298495,252820047
%N A351638 Number of length n word structures with all distinct run-lengths using an infinite alphabet.
%C A351638 Permuting the symbols will not change the structure.
%C A351638 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.
%H A351638 Andrew Howroyd, <a href="/A351638/b351638.txt">Table of n, a(n) for n = 0..500</a>
%e A351638 The a(3) = 3 words are 111, 112, 122.
%e A351638 The a(4) = 3 words are 1111, 1112, 1222. The word 1122 is not included because both runs have the same length.
%e A351638 The a(6) = 17 words are 111111, 111112, 111122, 111211, 111221, 112111, 112221, 112222, 122111, 122211, 122222, 111223, 111233, 112333, 112223, 122333, 122233.
%o A351638 (PARI)
%o A351638 P(n) = {Vec(-1 + prod(k=1, n, 1 + y*x^k + O(x*x^n)))}
%o A351638 R(u, k) = {k*[subst(serlaplace(p)/y, y, k-1) | p<-u]}
%o A351638 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!) ))}
%Y A351638 Row sums of A351637.
%Y A351638 Cf. A351292, A351642.
%K A351638 nonn
%O A351638 0,4
%A A351638 _Andrew Howroyd_, Feb 15 2022