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E.g.f. satisfies A(x) = 1 - x*log(1 - x*A(x)).
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%I #12 Mar 11 2024 05:35:42

%S 1,0,2,3,32,210,2184,26460,373344,6150816,113958720,2362345920,

%T 54094694400,1355708296800,36926213869440,1085886303989760,

%U 34291129916574720,1157362522046277120,41576054625791078400,1583864892141097098240,63779322541075124428800

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

%F a(n) = n! * Sum_{k=0..floor(n/2)} |Stirling1(n-k,k)|/(n-2*k+1)!.

%F a(n) ~ sqrt(2 - r*(2*r+1)) * n^(n-1) / (exp(n) * r^n), where r = 0.4599065470184992266076522060382204730855199647380... is the root of the equation 1/r + 2*r*log(r) = 1+r. - _Vaclav Kotesovec_, Mar 11 2024

%t nmax = 20; A[_] = 0; Do[A[x_] = 1 - x*Log[1 - x*A[x]] + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x] * Range[0,nmax]! (* _Vaclav Kotesovec_, Mar 11 2024 *)

%o (PARI) a(n) = n!*sum(k=0, n\2, abs(stirling(n-k, k, 1))/(n-2*k+1)!);

%Y Cf. A138013, A371118.

%Y Cf. A371115.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Mar 11 2024