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%I #5 Jul 17 2020 00:35:45
%S 1,1,5,55,1054,31046,1299386,73211510,5338080280,488727800664,
%T 54865512897432,7408400404206792,1184230737883333680,
%U 221121985937352261360,47683177920267470877648,11758982455716373002624816,3287966057434181416523799936
%N a(0) = 1; a(n) = n! * Sum_{k=1..n} binomial(n-1,k-1) * (k-1)! * H(k) * a(n-k) / (n-k)!, where H(k) is the k-th harmonic number.
%F Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(Sum_{n>=1} H(n) * x^n / n).
%F Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(log(1 - x)^2 / 2 + polylog(2,x)).
%t a[0] = 1; a[n_] := a[n] = n! Sum[Binomial[n - 1, k - 1] (k - 1)! HarmonicNumber[k] a[n - k]/(n - k)!, {k, 1, n}]; Table[a[n], {n, 0, 16}]
%t nmax = 16; CoefficientList[Series[Exp[Sum[HarmonicNumber[k] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!^2
%t nmax = 16; CoefficientList[Series[Exp[Log[1 - x]^2/2 + PolyLog[2, x]], {x, 0, nmax}], x] Range[0, nmax]!^2
%Y Cf. A001008, A002805, A074707, A087761, A235385, A235685, A336290.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jul 16 2020
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