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E.g.f. satisfies A(x) = 1/(1 - x*exp(x*A(x))).
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%I #19 Oct 07 2024 01:06:29

%S 1,1,4,27,268,3525,57966,1146061,26500552,702069129,20974309210,

%T 697754762001,25584428686620,1025230366195789,44579963354153878,

%U 2090676600895922565,105191995364927688976,5652501986238910061073,323083811850594613809714,19573120681427758058921881

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

%H Seiichi Manyama, <a href="/A161633/b161633.txt">Table of n, a(n) for n = 0..374</a>

%F E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! satisfies the following formulas.

%F (1) A(x) = 1 + x*A(x)*exp(x*A(x)).

%F (2) A(x) = (1/x) * Series_Reversion( x/(1 + x*exp(x)) ).

%F (3) A(x) = 1 + (m+1) * Sum{n>=1} n*(n+m)^(n-2) * x^n/n! * A(x)^n * exp(-(n+m-1)*x*A(x)) for all fixed nonnegative m.

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

%F Let A(x)^m = Sum_{n>=0} a(n,m)*x^n/n! then a(n,m) = n! * Sum_{k=0..n} binomial(n+m,k)*m/(n+m) * k^(n-k)/(n-k)!.

%F a(n) ~ n^(n-1) * c * ((c-1)*c)^(n+1/2) / (sqrt(2*c-1) * exp(n)), where c = 1 + 1/(2*LambertW(1/2)) = 2.4215299358831166... - _Vaclav Kotesovec_, Jan 10 2014

%e E.g.f.: A(x) = 1 + x + 4*x^2/2! + 27*x^3/3! + 268*x^4/4! + 3525*x^5/5! +...

%e exp(x*A(x)) = 1 + x + 3*x^2/2! + 19*x^3/3! + 181*x^4/4! + 2321*x^5/5! +...

%t Flatten[{1,Table[n!*Sum[Binomial[n+1,k]/(n+1) * k^(n-k)/(n-k)!,{k,0,n}],{n,1,20}]}] (* _Vaclav Kotesovec_, Jan 10 2014 *)

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

%Y Cf. A006153, A161630 (e.g.f. = exp(x*A(x))), A213644, A364980, A364981.

%Y Cf. A366232, A366233, A366234, A366235.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Jun 18 2009