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A138293 E.g.f. satisfies: A(x) = x + exp(x*A(x)).

%I A138293 #9 May 03 2023 05:54:05
%S A138293 1,2,5,28,245,2816,40537,702976,14270153,332102656,8719631981,
%T A138293 255020847104,8222803663549,289815184113664,11085650268060929,
%U A138293 457386463819595776,20248713707077863953,957435459515190345728
%N A138293 E.g.f. satisfies: A(x) = x + exp(x*A(x)).
%F A138293 a(n) = n!*Sum_{k=0..floor((n+1)/2)} (n-2*k+1)^(n-k)/(k!*(n-2*k+1)!). - _Vladeta Jovovic_, Mar 15 2008
%F A138293 a(n) ~ sqrt(1 + LambertW(2*exp(-2))) * (2/LambertW(2*exp(-2)))^((n+1)/2) * n^(n-1) / exp(n). - _Vaclav Kotesovec_, Nov 21 2017
%e A138293 E.g.f: A(x) = 1 + 2*x + 5/2*x^2 + 14/3*x^3 + 245/24*x^4 + 352/15*x^5 +...
%e A138293 Log(A(x)-x) = x + 2*x^2 + 5/2*x^3 + 14/3*x^4 + 245/24*x^5 + 352/15*x^6 +...
%p A138293 A138293 := proc(n)
%p A138293     add( (n-2*k+1)^(n-k)/k!/(n-2*k+1)!,k=0..(n+1)/2) ;
%p A138293     %*n! ;
%p A138293 end proc:
%p A138293 seq(A138293(n),n=0..30) ; # _R. J. Mathar_, May 03 2023
%t A138293 nmax = 20; CoefficientList[Series[x - ProductLog[-x*E^(x^2)]/x, {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Nov 21 2017 *)
%o A138293 (PARI) {a(n)=local(A=1);for(i=0,n,A=x+exp(x*A+x*O(x^n)));n!*polcoeff(A,n)}
%K A138293 nonn
%O A138293 0,2
%A A138293 _Paul D. Hanna_, Mar 13 2008