cp's OEIS Frontend

A351732 Number of length n word structures using an infinite alphabet with all distinct run-lengths and the first run length of a symbol less than that of previous symbols.

%I A351732 #8 Jan 28 2023 22:07:39
%S A351732 1,1,1,2,2,3,7,8,12,17,46,51,84,114,172,437,520,810,1153,1699,2298,
%T A351732 6075,6955,11219,15561,23308,31133,45544,107379,128475,200201,281480,
%U A351732 413389,561028,806643,1071165,2514418,2952086,4619012,6364285,9436458
%N A351732 Number of length n word structures using an infinite alphabet with all distinct run-lengths and the first run length of a symbol less than that of previous symbols.
%C A351732 Permuting the symbols does not change the structure.
%H A351732 Andrew Howroyd, <a href="/A351732/b351732.txt">Table of n, a(n) for n = 0..500</a>
%e A351732 The a(3) = 2 word structures are 111, 112.
%e A351732 The a(4) = 2 word structures are 1111, 1112.
%e A351732 The a(5) = 3 word structures are 11111, 11112, 11122.
%e A351732 The a(6) = 7 word structures are 111111, 111112, 111122, 111221, 111211, 112111, 111223.
%o A351732 (PARI)
%o A351732 P(n) = {Vec(-1 + prod(k=1, n, 1 + y*x^k + O(x*x^n)))}
%o A351732 R(u, k) = {k*[subst(serlaplace(p)/y, y, k-1) | p<-u]}
%o A351732 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!)^2) ))}
%Y A351732 Row sums of A351645.
%Y A351732 Cf. A351638.
%K A351732 nonn
%O A351732 0,4
%A A351732 _Andrew Howroyd_, May 20 2022