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%I #5 Jul 13 2021 16:58:44
%S 1,-1,-1,5,28,724,36,220716,-1255680,110979072,2530310400,
%T 1193835283200,-24457819622400,21656019855744000,899271273253248000,
%U 474367063601421849600,45822442913828595302400,28365278076547150440038400,2614371018285307258994688000
%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 * ( Sum_{d|k} 1 / (k/d)^(2*d-1) ) * a(n-k).
%t nmax = 18; 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 Sum[1/(k/d)^(2 d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 18}]
%Y Cf. A249588, A292359, A326864, A346313.
%K sign
%O 0,4
%A _Ilya Gutkovskiy_, Jul 13 2021