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A[n_, k_] := A[n, k] = If[n == 0, 1, k/n*
Sum[Binomial[2*n, j]*(n-j) *If[j == 0, 1, (k - 1)^j], {j, 0, n - 1}]];
T[n_, k_] := T[n, k] =
Sum[A[n, k - i]*(-1)^i*Binomial[k, i], {i, 0, k}]/If[k == 0, 1, k];
a[n_] := Sum[T[n, k], {k, 0, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 19 2022, after Alois P. Heinz *)
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a(n) = Sum_{k=0..n} A256116(n,k).
A:= proc(n, k) option remember; `if`(n=0, 1, k/n*
add(binomial(2*n, j) *(n-j) *(k-1)^j, j=0..n-1))
end:
T:= proc(n, k) option remember;
add(A(n, k-i)*(-1)^i*binomial(k, i), i=0..k)/`if`(k=0, 1, k)
end:
a:= n-> add(T(n, k), k=0..n):
seq(a(n), n=0..20);
a(3) = 20: aaaaaa, aaaabb, aaabba, aabaab, aabbaa, aabbbb, aabbcc, aabccb, aacbbc, aaccbb, abaaba, abbaaa, abbabb, abbacc, abbbba, abbcca, abccba, acbbca, accabb, accbba.
a(0) = 1: the empty word.
a(1) = 1: aa.
a(2) = 3: aaaa, aabb, abba.
a
Number of words of semilength n over n-ary alphabet, either empty or beginning with the first letter of the alphabet, such that the index set of occurring letters is an integer interval [1, k], that can be built by repeatedly inserting doublets into the initially empty word.