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E.g.f. satisfies A(x) = exp(x * (1 + x + x^2) * A(x^3/6)).
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%I #7 Nov 29 2023 11:47:59

%S 1,1,3,13,53,301,1951,13203,105673,919873,8472491,86799241,948033373,

%T 10924180853,135880443063,1780842778471,24496224075921,

%U 357483642165313,5454904465819603,86909842633518373,1453042115780967941,25262405474642837341

%N E.g.f. satisfies A(x) = exp(x * (1 + x + x^2) * A(x^3/6)).

%F a(0) = 1; a(n) = (n-1)! * Sum_{k=0..n-1} (k+1) * a(floor(k/3)) * a(n-1-k) / (6^floor(k/3) * floor(k/3)! * (n-1-k)!).

%o (PARI) a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=0, i-1, (j+1)*v[j\3+1]*v[i-j]/(6^(j\3)*(j\3)!*(i-1-j)!))); v;

%Y Cf. A367754, A367756, A367757.

%Y Cf. A143567.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Nov 29 2023