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E.g.f. satisfies A(x) = 1 - log(1 - x*A(x)^3).
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%I #10 Nov 07 2023 08:56:39

%S 1,1,7,101,2250,68184,2619822,122071704,6689791392,421670267136,

%T 30055781201520,2390512621714656,209893714832795760,

%U 20165895195283566000,2104433775967024226592,237043144515185017456320,28664975599576485530851584,3704019298858867019823244800

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

%F a(n) = (3*n)! * Sum_{k=0..n} |Stirling1(n,k)|/(3*n-k+1)!.

%F a(n) ~ (-3 - LambertW(-1, -3*exp(-4)))^(2*n+1) * (-LambertW(-1, -3*exp(-4)))^n * n^(n-1) / (sqrt(-3 - 3*LambertW(-1, -3*exp(-4))) * exp(n)). - _Vaclav Kotesovec_, Nov 07 2023

%t Table[(3*n)! * Sum[Abs[StirlingS1[n,k]]/(3*n-k+1)!, {k,0,n}], {n,0,20}] (* _Vaclav Kotesovec_, Nov 07 2023 *)

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

%Y Cf. A138013, A367080.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Nov 07 2023