%I #4 Jul 13 2021 16:58:50

%S 1,1,3,31,496,12576,444736,22056448,1406058816,114618828096,

%T 11405077216704,1385889578069184,198961869847145472,

%U 33725910553646229504,6594186368339077238784,1487133154121568112705536,379990326228614750079369216,110013397755650063836228435968

%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 * ( Sum_{d|k} (-1)^d / (k/d)^(2*d-1) ) * 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 Sum[(-1)^d/(k/d)^(2 d - 1), {d, Divisors[k]}] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 17}]

%Y Cf. A249588, A292358, A326864, A346312.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Jul 13 2021