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E.g.f. satisfies A(x) = 1 + x^3*exp(x*A(x)).
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%I #10 Mar 10 2024 08:19:49

%S 1,0,0,6,24,60,120,5250,80976,726264,4839120,86487390,2283242280,

%T 42585905076,590667519624,10115535833130,286758920451360,

%U 8128299117822960,186279550983756576,4123388294626654134,118916807955913504440,4102548791571529697580

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

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

%t nmax = 20; CoefficientList[Series[1 - LambertW[-E^x*x^4]/x, {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Mar 10 2024 *)

%o (PARI) a(n) = n!*sum(k=0, n\3, k^(n-3*k)*binomial(n-3*k+1, k)/((n-3*k+1)*(n-3*k)!));

%Y Cf. A370985, A371019, A371045, A371046.

%Y Cf. A161631, A371042.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Mar 09 2024