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A177380 E.g.f. satisfies: A(x) = 1+x + x*log(A(x)).

%I A177380 #9 Feb 24 2014 17:08:00
%S A177380 1,1,2,3,-4,-50,-36,2058,10800,-131616,-1975680,7741800,417480480,
%T A177380 1307617584,-101626746144,-1284067345680,25419094122240,
%U A177380 791333924647680,-3900043588999680,-472446912421801728,-3183064994777932800
%N A177380 E.g.f. satisfies: A(x) = 1+x + x*log(A(x)).
%C A177380 The signs have a complex structure; are they periodic after some point?
%H A177380 Vaclav Kotesovec, <a href="/A177380/b177380.txt">Table of n, a(n) for n = 0..420</a>
%F A177380 E.g.f.: A(x) = 1 + Series_Reversion( x/(1 + log(1+x)) ).
%F A177380 ...
%F A177380 Let G(x) = e.g.f. of A138013, then G(x) and A(x) satisfy:
%F A177380 (1) [x^n] A(x)^n = [x^(n+1)] A(x)^n = A138013(n)/(n-1)! for n>=1;
%F A177380 (2) A(x/(1 - x*G(x))) = 1/(1 - x*G(x));
%F A177380 (3) G(x) = 1 - log(1 - x*G(x)) = Series_Reversion(x/(1-log(1-x)))/x.
%F A177380 ...
%F A177380 Let F(x) = e.g.f. of A177379, then F(x) and A(x) satisfy:
%F A177380 (4) [x^n] A(x)^(n+1)/(n+1) = A177379(n)/n! for n>=0;
%F A177380 (5) A(x*F(x)) = F(x) and F(x/A(x)) = A(x);
%F A177380 (6) F(x) = 1/(1 - x*G(x)) = 1/(1 - Series_Reversion(x/(1-log(1-x)))).
%F A177380 Lim sup n->infinity (|a(n)|/n!)^(1/n) = abs(LambertW(-1)) = 1.3745570107437... (see A238274). - _Vaclav Kotesovec_, Jan 11 2014
%e A177380 E.g.f: A(x) = 1 + x + 2*x^2/2! + 3*x^3/3! - 4*x^4/4! - 50*x^5/5! +...
%e A177380 log(A(x)) = 2*x/2! + 3*x^2/3! - 4*x^3/4! - 50*x^4/5! - 36*x^5/5! +...
%e A177380 ...
%e A177380 Coefficients in the initial powers of A(x) begin:
%e A177380 [1,(1),(1), 1/2, -1/6, -5/12, -1/20, 49/120, 15/56, -457/1260,...];
%e A177380 [1, 2,(3),(3), 5/3, -1/6, -61/60, -17/60, 272/315, 451/630,...];
%e A177380 [1, 3, 6,(17/2),(17/2), 21/4, 3/5, -83/40, -187/168, 115/84,...];
%e A177380 [1, 4, 10, 18,(73/3),(73/3), 163/10, 131/30, -261/70, -1093/315,...];
%e A177380 [1, 5, 15, 65/2, 325/6,(847/12),(847/12), 1205/24, 9551/504,...];
%e A177380 [1, 6, 21, 53, 104, 327/2,(4139/20),(4139/20), 6469/42, 7414/105,...];
%e A177380 [1, 7, 28, 161/2, 1085/6, 3955/12, 4949/10,(24477/40),(24477/40),...];
%e A177380 [1, 8, 36, 116, 878/3, 1810/3, 15569/15, 7509/5,(114760/63),(114760/63), ...]; ...
%e A177380 where the coefficients in parenthesis illustrate the property
%e A177380 that the coefficients of x^n and x^(n+1) in A(x)^n are equal:
%e A177380 [x^n] A(x)^n = [x^(n+1)] A(x)^n = A138013(n)/(n-1)!,
%e A177380 where G(x) = e.g.f. of A138013 begins:
%e A177380 G(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 146*x^4/4! + 1694*x^5/5! + ...
%e A177380 and satisfies: exp(1 - G(x)) = 1 - x*G(x).
%t A177380 CoefficientList[1+InverseSeries[Series[x/(1 + Log[1+x]), {x, 0, 20}], x],x] * Range[0, 20]! (* _Vaclav Kotesovec_, Jan 11 2014 *)
%o A177380 (PARI) {a(n)=n!*polcoeff(1+serreverse(x/(1+log(1+x+x*O(x^n)))),n)}
%o A177380 (PARI) {a(n)=local(A=1+x);for(i=1,n,A=1+x+x*log(A+O(x^n)));n!*polcoeff(A,n)}
%Y A177380 Cf. A177379, A138013, A038037, A238274.
%K A177380 sign
%O A177380 0,3
%A A177380 _Paul D. Hanna_, May 14 2010