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%I #7 Jul 29 2020 16:58:01
%S 1,0,-3,-19,-75,574,25795,579963,9342529,21955076,-7954085799,
%T -535479422655,-25206613635203,-871888114433454,-7465407495946777,
%U 2538884115164554199,344689220434285963905,31689538033223254172648,2273498459548301881979029
%N Sum_{n>=0} a(n) * x^n / (n!)^2 = exp(x) * BesselI(0,2*sqrt(1 - exp(x))).
%F a(n) = n! * Sum_{k=0..n} (-1)^k * Stirling2(n+1,k+1) / k!.
%t nmax = 18; CoefficientList[Series[Exp[x] BesselI[0, 2 Sqrt[1 - Exp[x]]], {x, 0, nmax}], x] Range[0, nmax]!^2
%t Table[n! Sum[(-1)^k StirlingS2[n + 1, k + 1]/k!, {k, 0, n}], {n, 0, 18}]
%o (PARI) a(n) = n! * sum(k=0, n, (-1)^k*stirling(n+1,k+1,2) / k!); \\ _Michel Marcus_, Jul 29 2020
%Y Cf. A001569, A009940, A293037, A336588.
%K sign
%O 0,3
%A _Ilya Gutkovskiy_, Jul 26 2020