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%I #4 May 04 2024 15:00:46
%S 0,1,2,16,264,7296,302720,17587200,1362399360,135693537280,
%T 16893684928512,2570631845806080,469393033744588800,
%U 101294080603625226240,25502237392032633323520,7408331513180811911233536,2459543337577081650719784960,925435622656059412145504256000
%N Sum_{n>=0} a(n) * x^n / (n!)^2 = -log(BesselJ(0,2*sqrt(2*x))) / 2.
%F a(0) = 0; a(n) = (-2)^(n-1) - (1/n) * Sum_{k=1..n-1} binomial(n,k)^2 * (-2)^k * (n-k) * a(n-k).
%F a(n) = 2^(n-1) * A002190(n).
%t nmax = 17; CoefficientList[Series[-Log[BesselJ[0, 2 Sqrt[2 x]]]/2, {x, 0, nmax}], x] Range[0, nmax]!^2
%t a[0] = 0; a[n_] := a[n] = (-2)^(n - 1) - (1/n) Sum[Binomial[n, k]^2 (-2)^k (n - k) a[n - k], {k, 1, n - 1}]; Table[a[n], {n, 0, 17}]
%Y Cf. A002190.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, May 04 2024
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