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%I #5 Jul 12 2021 18:01:32
%S 1,1,2,7,41,346,3807,53747,952275,20362552,515112983,15277888693,
%T 523304644304,20415373547609,900219731675981,44533809102813206,
%U 2451041479421900803,149140880201760643360,9982798939295116151967,731215136812226462200109,58333374310397488522052976
%N Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( BesselI(0,2*sqrt(x)) - 1 - x^2 / 4 ).
%F Sum_{n>=0} a(n) * x^n / (n!)^2 = exp( x + Sum_{n>=3} x^n / (n!)^2 ).
%F a(0) = 1; a(n) = n * a(n-1) + (1/n) * Sum_{k=3..n} binomial(n,k)^2 * k * a(n-k).
%t nmax = 20; CoefficientList[Series[Exp[BesselI[0, 2 Sqrt[x]] - 1 - x^2/4], {x, 0, nmax}], x] Range[0, nmax]!^2
%t a[0] = 1; a[n_] := a[n] = n a[n - 1] + (1/n) Sum[Binomial[n, k]^2 k a[n - k], {k, 3, n}]; Table[a[n], {n, 0, 20}]
%Y Cf. A023998, A061696, A097514, A346272.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, Jul 12 2021
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