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%I #27 Feb 28 2023 06:48:51
%S 1,1,3,4,-19,-64,1207,5440,-164071,-954368,39943691,284754944,
%T -15250391099,-128749666304,8402599565375,81978198409216,
%U -6309988001033167,-69853770233675776,6194681665486634899,76717804389440684032
%N E.g.f. satisfies A(x) = exp(x/A(-x)).
%H Seiichi Manyama, <a href="/A196198/b196198.txt">Table of n, a(n) for n = 0..406</a>
%F a(n) = Sum_{k=0..n-1} binomial(n,k) * (n-k)^k * (-k+1)^(n-k-1) for n>0 with a(0)=1.
%F E.g.f. satisfies:
%F _ A(x) = exp(x*exp(x/A(x))).
%F _ A(x) = exp(x* exp(x*exp(-x*exp(x*exp(-x*exp(x*exp(-x*...))))))).
%F _ A(x) = exp(x*B(x)) where B(x) = exp(x/B(x)) is the e.g.f. of A141369.
%F E.g.f. satisfies: x/exp(-x/A(x)) = log(A(x)). - _Vaclav Kotesovec_, Feb 26 2014
%F |a(n)| ~ c * n! / (n^(3/2) * r^n), where r = 0.5098636055230131449434409623392631606695606770070519241... is the root of the equation r*exp(1/LambertW(-I/r))/I = LambertW(-I/r), and c = 0.385745347287849929987791864025522098993432068... if n is even, and c = 0.12921599603996711137996765405025929272341118... if n is odd. - _Vaclav Kotesovec_, Feb 26 2014
%e E.g.f.: A(x) = 1 + x + 3*x^2/2! + 4*x^3/3! - 19*x^4/4! - 64*x^5/5! +...
%e where log(A(x)) = x/A(-x) begins:
%e x/A(-x) = x + 2*x^2/2! - 3*x^3/3! - 32*x^4/4! + 105*x^5/5! + 2016*x^6/6! - 10115*x^7/7! - 282624*x^8/8! +...+ n*A141369(n-1)*x^n/n! +...
%t Flatten[{1,1,3,Table[Sum[Binomial[n,k]*(n-k)^k*(-k+1)^(n-k-1),{k,0,n-1}],{n,3,20}]}] (* _Vaclav Kotesovec_, Feb 26 2014 *)
%o (PARI) {a(n)=if(n==0,1,sum(k=0, n-1, binomial(n, k)*(n-k)^k*(-k+1)^(n-k-1)))}
%o (PARI) {a(n)=local(A=1+x);for(i=1,n,A=exp(x/subst(A,x,-x+x*O(x^n))));n!*polcoeff(A,n)}
%Y Cf. A141369.
%K sign
%O 0,3
%A _Paul D. Hanna_, Sep 30 2011
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