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%I #52 Jan 23 2021 17:33:08
%S 0,1,3,28,447,11176,402335,19714416,1261722623,102199532464,
%T 10219953246399,1236614342814280,178072465365256319,
%U 30094246646728317912,5898472342758750310751,1327156277120718819918976,339752006942904017899257855,98188330006499261172885520096
%N a(n) = (n!)^2 * Sum_{k=1..n} (-1)^(k+1) / (k!)^2.
%F Sum_{n>=0} a(n) * x^n / (n!)^2 = (1 - BesselJ(0,2*sqrt(x))) / (1 - x).
%F a(0) = 0; a(n) = n^2 * a(n-1) - (-1)^n.
%t Table[n!^2 Sum[(-1)^(k + 1)/k!^2, {k, 1, n}], {n, 0, 17}]
%t nmax = 17; CoefficientList[Series[(1 - BesselJ[0, 2 Sqrt[x]])/(1 - x), {x, 0, nmax}], x] Range[0, nmax]!^2
%Y Cf. A002467, A006040, A066998, A073701.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jan 23 2021