%I #6 Feb 15 2021 22:36:50
%S 1,0,4,5,58,217,2035,13470,134164,1243770,14129410,164244808,
%T 2151576620,29671566836,444758323628,7055358559376,119546765395744,
%U 2139179551573104,40486788832168944,805969129348431936,16860672502118423136,369459637224850523808,8467140450141232328160
%N E.g.f.: log(1 - log(1 - x))^2 / 2.
%F a(n) = Sum_{k=2..n} |Stirling1(n, k)| * Stirling1(k, 2).
%F a(n) = (-1)^n * Sum_{k=2..n} Stirling1(n, k) * (k-1)! * H(k-1), where H(k) is the k-th harmonic number.
%t nmax = 24; CoefficientList[Series[Log[1 - Log[1 - x]]^2/2, {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 2] &
%t Table[Sum[Abs[StirlingS1[n, k]] StirlingS1[k, 2], {k, 2, n}], {n, 2, 24}]
%Y Cf. A000254, A000558, A008275, A079642, A081048, A089064, A302547, A302548, A341587.
%K nonn
%O 2,3
%A _Ilya Gutkovskiy_, Feb 15 2021