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%I #4 Aug 24 2023 10:33:59
%S 1,2,11,107,1577,32201,860460,28921567,1187475909,58232016701,
%T 3350187053856,222857979706305,16935374386652282,1455271176236200143,
%U 140181486948923188907,15023106134895469195114,1779460642743292348315607,231607462899834684300774917,32954119475274480307491604062,5102159139278049158548905019487
%N Sum_{n>=0} a(n) * x^n / n!^2 = exp( Sum_{n>=1} prime(n) * x^n / n!^2 ).
%F a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} binomial(n,k)^2 * k * prime(k) * a(n-k).
%t nmax = 19; CoefficientList[Series[Exp[Sum[Prime[k] 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]^2 k Prime[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 19}]
%Y Cf. A007446, A023998.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Aug 21 2023