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A361777 Expansion of e.g.f. A(x) satisfying A(x) = exp( x * A(x)^x ).

%I A361777 #32 May 03 2023 04:37:51
%S A361777 1,1,1,7,25,241,1561,19951,188497,3032065,37720081,734331511,
%T A361777 11341504681,259658249137,4792613587945,126280535523871,
%U A361777 2712093428032801,80881163134899841,1981706113050012577,66009436508505875815,1817280748378601067961,66887742743997848317681
%N A361777 Expansion of e.g.f. A(x) satisfying A(x) = exp( x * A(x)^x ).
%C A361777 Radius of convergence of e.g.f. A(x) is r = exp(-1/2) = 0.6065306597126334...; at x = r, A(x) converges: A( exp(-1/2) ) = exp( exp(1/2) ) = 5.200325764789961...
%H A361777 Seiichi Manyama, <a href="/A361777/b361777.txt">Table of n, a(n) for n = 0..416</a>
%F A361777 E.g.f. A(x) = Sum_{n>=0} a(n)*x^n/n! may be defined by the following.
%F A361777 (1) A(x) = exp( x * A(x)^x ).
%F A361777 (2) A(x) = ( LambertW(-x^2)/(-x^2) )^(1/x).
%F A361777 (3) A(x) = Sum_{n>=0} (1 + n*x)^(n-1) * x^n/n!.
%F A361777 (4) A(x) = exp( -LambertW(-x^2) / x ).
%F A361777 (5) A(x) = exp( Sum_{n>=1} n^(n-1) * x^(2*n-1) / n! ).
%F A361777 (6) A(x) = 1 + Sum_{n>=1} x^n/n! * Sum_{k>=0} n*(n+k)^(k-1) * x^(2*k) / k!.
%F A361777 (7) a(n) = Sum_{k=0..floor(n/2)} (n-2*k)*(n-k)^(k-1) * n!/((n-2*k)!*k!) for n > 0, with a(0) = 1.
%F A361777 (8) A'(x) = A(x)^(1+x) * (1 + x^2*A(x)^x) / (1 - x^2*A(x)^x).
%F A361777 a(n) ~ sqrt(2) * (exp(2*exp(1/2)) - (-1)^n) * n^(n-1) / exp((n-1)/2 + exp(1/2)). - _Vaclav Kotesovec_, Apr 17 2023
%e A361777 E.g.f.: A(x) = 1 + x + x^2/2! + 7*x^3/3! + 25*x^4/4! + 241*x^5/5! + 1561*x^6/6! + 19951*x^7/7! + 188497*x^8/8! + 3032065*x^9/9! + 37720081*x^10/10! + ...
%e A361777 where
%e A361777 A(x) = exp( x * A(x)^x )
%e A361777 and
%e A361777 A(x) = exp(x + 2*x^3/2! + 3^2*x^5/3! + 4^3*x^7/4! + 5^4*x^9/5! + 6^5*x^11/6! + 7^6*x^13/7! + 8^7*x^15/8! + ... + n^(n-1)*x^(2*n-1)/n! + ...).
%e A361777 Also,
%e A361777 A(x) = 1 + x + (1+2*x)*x^2/2! + (1+3*x)^2*x^3/3! + (1+4*x)^3*x^4/4! + (1+5*x)^4*x^5/5! + (1+6*x)^5*x^6/6! + ... + (1 + n*x)^(n-1)*x^n/n! + ...
%e A361777 SPECIFIC VALUES.
%e A361777 A( exp(-1/2) ) = exp( exp(1/2) ) = 5.2003257647899611368422256424148878...
%t A361777 nmax = 20; CoefficientList[Series[E^(-LambertW[-x^2]/x), {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Apr 17 2023 *)
%o A361777 (PARI) {a(n) = my(A=1); for(i=1,n, A = exp( x*A^x +x*O(x^n) ) ); n!*polcoeff(A,n)}
%o A361777 for(n=0,25,print1(a(n),", "))
%o A361777 (PARI) {a(n) = if(n==0,1, sum(k=0,n\2, (n-2*k)*(n-k)^(k-1) * n!/((n-2*k)!*k!) ) )}
%o A361777 for(n=0,25,print1(a(n),", "))
%o A361777 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(x*exp(-lambertw(-x^2))))) \\ _Seiichi Manyama_, Apr 25 2023
%Y A361777 Cf. A000272.
%K A361777 nonn
%O A361777 0,4
%A A361777 _Paul D. Hanna_, Apr 17 2023