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%I #7 Jul 17 2020 03:49:37
%S 0,1,-2,3,8,305,10734,502747,30344992,2307890097,216571514030,
%T 24619605092291,3337294343698248,532148381719443073,
%U 98646472269855762238,21041945289232131607995,5118447176652195630775424,1408601897794844346184122017,435481794298015565250651718302
%N a(n) = (n!)^2 * Sum_{k=1..n} (-1)^(n-k) / (k * ((n-k)!)^2).
%F Sum_{n>=0} a(n) * x^n / (n!)^2 = -log(1 - x) * BesselJ(0,2*sqrt(x)).
%t Table[(n!)^2 Sum[(-1)^(n - k)/(k ((n - k)!)^2), {k, 1, n}], {n, 0, 18}]
%t nmax = 18; CoefficientList[Series[-Log[1 - x] BesselJ[0, 2 Sqrt[x]], {x, 0, nmax}], x] Range[0, nmax]!^2
%o (PARI) a(n) = (n!)^2 * sum(k=1, n, (-1)^(n-k) / (k * ((n-k)!)^2)); \\ _Michel Marcus_, Jul 17 2020
%Y Cf. A002741, A009940, A336291.
%K sign
%O 0,3
%A _Ilya Gutkovskiy_, Jul 16 2020