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%I #4 Sep 02 2020 19:24:15
%S 0,1,0,-3,28,-215,-174,90223,-3840472,103719537,429704110,
%T -357346077869,35100093531900,-2005608652057595,-24108041118593418,
%U 27881407632242902515,-4876442148527153942384,474102062424164433715937,12637408141631813073125094,-18867461801192524662360616421
%N a(0) = 0; a(n) = n - (1/n) * Sum_{k=1..n-1} binomial(n,k)^2 * (n-k) * k * a(k).
%F Sum_{n>=0} a(n) * x^n / (n!)^2 = log(1 + sqrt(x) * BesselI(1,2*sqrt(x))).
%F Sum_{n>=0} a(n) * x^n / (n!)^2 = log(1 + Sum_{n>=1} n * x^n / (n!)^2).
%t a[0] = 0; a[n_] := a[n] = n - (1/n) Sum[Binomial[n, k]^2 (n - k) k a[k], {k, 1, n - 1}]; Table[a[n], {n, 0, 19}]
%t nmax = 19; CoefficientList[Series[Log[1 + Sqrt[x] BesselI[1, 2 Sqrt[x]]], {x, 0, nmax}], x] Range[0, nmax]!^2
%Y Cf. A002190, A009306, A336227.
%K sign
%O 0,4
%A _Ilya Gutkovskiy_, Sep 02 2020
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