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A351644 Number of length n word structures with all distinct runs using at most 3 symbols.

%I A351644 #8 Jan 28 2023 22:07:43
%S A351644 1,1,2,4,9,21,46,108,223,487,1028,2060,4327,8591,16818,33562,64441,
%T A351644 122983,232378,443446,816371,1503517,2775372,5052186,9116047,16231929,
%U A351644 29182198,51503788,91032821,158301653,275776810,479642780,824964483,1414293391,2403093256,4095230980
%N A351644 Number of length n word structures with all distinct runs using at most 3 symbols.
%C A351644 Permuting the symbols will not change the structure.
%H A351644 Andrew Howroyd, <a href="/A351644/b351644.txt">Table of n, a(n) for n = 0..1000</a>
%F A351644 a(n) = A351018(n) + A351643(n).
%F A351644 a(n) = Sum_{k=0..3} A351641(n,k).
%e A351644 The a(1) = 1 word is 1.
%e A351644 The a(2) = 2 words are 11, 12.
%e A351644 The a(3) = 4 words are 111, 112, 122, 123.
%e A351644 The a(4) = 9 words are 1111, 1112, 1121, 1122, 1211, 1222, 1123, 1223, 1233.
%o A351644 (PARI) \\ See A351641 for R, S.
%o A351644 seq(n)={my(q=S(n), c=3); concat([1], sum(k=1, c, R(q^k-1)*sum(r=k, c, binomial(r, k)*(-1)^(r-k)/r!) )); }
%Y A351644 Cf. A351018, A351641, A351643.
%K A351644 nonn
%O A351644 0,3
%A A351644 _Andrew Howroyd_, Feb 16 2022