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E.g.f. satisfies A(x) = 1 + x*A(x)*exp(x^2*A(x)).
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%I #12 Aug 31 2023 07:46:19

%S 1,1,2,12,96,900,10800,157080,2634240,50455440,1089849600,26157479040,

%T 690848040960,19924295751360,623024501299200,20996216063222400,

%U 758724126031872000,29267547577396128000,1200407895406514995200

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

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

%F a(n) ~ sqrt((s+1)/(2*s-1)) * (s-1)^((n+1)/2) * s^(n/2 + 1) * n^(n-1) / exp(n), where s = 3.011547791499065828694160466323712196300874261862... is the root of the equation (s-1)*LambertW(2*(s-1)^2/s) = 2. - _Vaclav Kotesovec_, Aug 31 2023

%t Join[{1}, Table[n! * Sum[(n-2*k)^k * Binomial[n-k+1,n-2*k] / ((n-k+1)*k!), {k,0,Floor[n/2]}], {n,1,20}]] (* _Vaclav Kotesovec_, Aug 31 2023 *)

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

%Y Cf. A358064, A365283, A365284.

%Y Cf. A365285.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Aug 31 2023