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%I #4 Jul 13 2021 16:59:02
%S 1,1,3,28,483,11976,423660,20801775,1337182819,108259612048,
%T 10814058518328,1308659192928495,188498906179378476,
%U 31855351764833425895,6243218508505581436249,1404734813476218805338303,359618310105650201828166499,103929494668760259335327432160
%N Sum_{n>=0} a(n) * x^n / (n!)^2 = Product_{n>=1} 1 / (1 + (-x)^n / (n!)^2).
%F a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} (-1)^k * (binomial(n,k) * k!)^2 * k * ( Sum_{d|k} (-1)^d / (d * ((k/d)!)^(2*d)) ) * a(n-k).
%t nmax = 17; CoefficientList[Series[Product[1/(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[(-1)^k (Binomial[n, k] k!)^2 k Sum[(-1)^d/(d ((k/d)!)^(2 d)), {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]
%Y Cf. A076901, A183229, A183240, A292308, A346314.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jul 13 2021