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A177384 G.f. satisfies: A(x) = 1 + x/(A(x) + x*A'(x)).

%I A177384 #11 Mar 06 2020 17:14:08
%S A177384 1,1,-2,10,-72,644,-6704,78408,-1008480,14065744,-210682080,
%T A177384 3365194560,-57019105920,1020662366400,-19238635678208,
%U A177384 380825404556288,-7898501807543808,171304216873595136,-3878189367387230720
%N A177384 G.f. satisfies: A(x) = 1 + x/(A(x) + x*A'(x)).
%H A177384 Vaclav Kotesovec, <a href="/A177384/b177384.txt">Table of n, a(n) for n = 0..440</a>
%F A177384 G.f. satisfies: d/dx x*A(x) = -x + (1/x)*d/dx { [x*A(x)]^2/2 }.
%F A177384 Let F(x) be the g.f. of A177383, then
%F A177384 . a(n) = [x^n] F(x)^(n+1)/(n+1);
%F A177384 . A(x) = F(x*A(x)) so that A(x) = (1/x)*Series_Reversion(x/F(x))
%F A177384 where F(x) satisfies: [x^n] F(x)^(n+1) = [x^n] F(x)^(n+2).
%F A177384 a(n) ~ (-1)^(n+1) * c * n! * n^4, where c = 0.005428317993266202636748034138.... - _Vaclav Kotesovec_, Feb 22 2014
%F A177384 Recurrence: a(0) = a(1) = 1, a(n) = -Sum_{0<k<n} (k+1)*a(k)*a(n-k). - _Vladimir Reshetnikov_, Nov 14 2016
%F A177384 A177383(n) / a(n) ~ exp(1). - _Vaclav Kotesovec_, Mar 06 2020
%e A177384 G.f.: A(x) = 1 + x - 2*x^2 + 10*x^3 - 72*x^4 + 644*x^5 - 6704*x^6 +...
%e A177384 d/dx x*A(x) = 1 + 2*x - 6*x^2 + 40*x^3 - 360*x^4 + 3864*x^5 -...
%e A177384 d/dx [x*A(x)]^2/2 = x + 3*x^2 - 6*x^3 + 40*x^4 - 360*x^5 + 3864*x^6 -...
%t A177384 a[0] = a[1] = 1; a[n_] := a[n] = -Sum[(k+1) a[k] a[n-k], {k, 1, n-1}]; Table[a[n], {n, 0, 20}] (* _Vladimir Reshetnikov_, Nov 14 2016 *)
%o A177384 (PARI) {a(n)=local(G=1+x);for(i=1,n,G=1+x/(G+x*deriv(G)+x*O(x^n)));polcoeff(G,n)}
%Y A177384 Cf. A177383.
%K A177384 sign
%O A177384 0,3
%A A177384 _Paul D. Hanna_, May 15 2010