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%I #11 Jan 23 2025 08:31:27
%S 1,-3,8,-20,49,-189,1791,-21132,228306,-2274690,22190772,-230289696,
%T 2756380782,-38757988710,608149754538,-10057914084048,171037444641816,
%U -3000345245061048,55157102668064592,-1077263181846230400,22411300073192730360,-492846784406541548280
%N Expansion of e.g.f. log( 1 + log(1 + x)^2 / 2 ).
%F a(n) = Stirling1(n,2) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling1(n-k,2) * k * a(k).
%F a(n) = Sum_{k=1..floor(n/2)} (-1)^(k-1) * (2*k)! * Stirling1(n,2*k)/(k * 2^k). - _Seiichi Manyama_, Jan 23 2025
%t nmax = 23; CoefficientList[Series[Log[1 + Log[1 + x]^2/2], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 2] &
%t a[n_] := a[n] = StirlingS1[n, 2] - (1/n) Sum[Binomial[n, k] StirlingS1[n - k, 2] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 2, 23}]
%Y Cf. A000254, A003713, A081048, A346945, A346946, A346947.
%Y Cf. A346974.
%K sign,changed
%O 2,2
%A _Ilya Gutkovskiy_, Aug 08 2021