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%I #5 Jul 13 2021 16:58:56
%S 1,-1,-1,8,15,124,-3340,-9311,-102641,-1880812,150047424,692058289,
%T 8916106452,167039809897,7435628931289,-1381243302601067,
%U -9407162843960561,-165954439670564988,-3103870029424074136,-123659189880256295879,-10671656695397289496160
%N Sum_{n>=0} a(n) * x^n / (n!)^2 = Product_{n>=1} (1 - x^n / (n!)^2).
%F a(0) = 1; a(n) = -(1/n) * Sum_{k=1..n} (binomial(n,k) * k!)^2 * k * ( Sum_{d|k} 1 / (d * ((k/d)!)^(2*d)) ) * a(n-k).
%t nmax = 20; CoefficientList[Series[Product[(1 - x^k/(k!)^2), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!^2
%t a[0] = 1; a[n_] := a[n] = -(1/n) Sum[(Binomial[n, k] k!)^2 k Sum[1/(d ((k/d)!)^(2 d)), {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 20}]
%Y Cf. A183229, A183240, A185895, A346315.
%K sign
%O 0,4
%A _Ilya Gutkovskiy_, Jul 13 2021