%I #86 Aug 20 2021 05:47:25
%S 1,2,6,24,108,552,3144,19392,129168,920736,6958944,55582848,466824384,
%T 4104798336,37688879232,360236187648,3575154053376,36768528142848,
%U 391060780180992,4293782854170624,48597548604926976,566152604314232832,6780179847538722816,83375209195856216064
%N E.g.f.: exp(2 * (x + x^2 / 2 + x^3 / 3)).
%F D-finite recurrence: a(n) = 2 * (a(n-1) + (n-1) * a(n-2) + (n-1) * (n-2) * a(n-3)).
%F a(n) = Sum_{k=0..n} binomial(n,k) * A057693(k) * A057693(n-k).
%t nmax = 23; CoefficientList[Series[Exp[2 (x + x^2/2 + x^3/3)], {x, 0, nmax}], x] Range[0, nmax]!
%t a[0] = 1; a[1] = 2; a[2] = 6; a[n_] := a[n] = 2 (a[n - 1] + (n - 1) a[n - 2] + (n - 1) (n - 2) a[n - 3]); Table[a[n], {n, 0, 23}]
%o (PARI) my(x='x+O('x^30)); Vec(serlaplace(exp(2*(x + x^2/2 + x^3/3)))) \\ _Michel Marcus_, Nov 21 2020
%Y Cf. A000898, A057693.
%K nonn
%O 0,2
%A _Ilya Gutkovskiy_, Nov 20 2020