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E.g.f. satisfies: A(x) = x + exp(x*A(x)).
2

%I #9 May 03 2023 05:54:05

%S 1,2,5,28,245,2816,40537,702976,14270153,332102656,8719631981,

%T 255020847104,8222803663549,289815184113664,11085650268060929,

%U 457386463819595776,20248713707077863953,957435459515190345728

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

%F 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 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 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 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 := proc(n)

%p add( (n-2*k+1)^(n-k)/k!/(n-2*k+1)!,k=0..(n+1)/2) ;

%p %*n! ;

%p end proc:

%p seq(A138293(n),n=0..30) ; # _R. J. Mathar_, May 03 2023

%t nmax = 20; CoefficientList[Series[x - ProductLog[-x*E^(x^2)]/x, {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Nov 21 2017 *)

%o (PARI) {a(n)=local(A=1);for(i=0,n,A=x+exp(x*A+x*O(x^n)));n!*polcoeff(A,n)}

%K nonn

%O 0,2

%A _Paul D. Hanna_, Mar 13 2008