A177380 - OEIS

%I #9 Feb 24 2014 17:08:00

%S 1,1,2,3,-4,-50,-36,2058,10800,-131616,-1975680,7741800,417480480,

%T 1307617584,-101626746144,-1284067345680,25419094122240,

%U 791333924647680,-3900043588999680,-472446912421801728,-3183064994777932800

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

%C The signs have a complex structure; are they periodic after some point?

%H Vaclav Kotesovec, <a href="/A177380/b177380.txt">Table of n, a(n) for n = 0..420</a>

%F E.g.f.: A(x) = 1 + Series_Reversion( x/(1 + log(1+x)) ).

%F ...

%F Let G(x) = e.g.f. of A138013, then G(x) and A(x) satisfy:

%F (1) [x^n] A(x)^n = [x^(n+1)] A(x)^n = A138013(n)/(n-1)! for n>=1;

%F (2) A(x/(1 - x*G(x))) = 1/(1 - x*G(x));

%F (3) G(x) = 1 - log(1 - x*G(x)) = Series_Reversion(x/(1-log(1-x)))/x.

%F ...

%F Let F(x) = e.g.f. of A177379, then F(x) and A(x) satisfy:

%F (4) [x^n] A(x)^(n+1)/(n+1) = A177379(n)/n! for n>=0;

%F (5) A(x*F(x)) = F(x) and F(x/A(x)) = A(x);

%F (6) F(x) = 1/(1 - x*G(x)) = 1/(1 - Series_Reversion(x/(1-log(1-x)))).

%F Lim sup n->infinity (|a(n)|/n!)^(1/n) = abs(LambertW(-1)) = 1.3745570107437... (see A238274). - _Vaclav Kotesovec_, Jan 11 2014

%e E.g.f: A(x) = 1 + x + 2*x^2/2! + 3*x^3/3! - 4*x^4/4! - 50*x^5/5! +...

%e log(A(x)) = 2*x/2! + 3*x^2/3! - 4*x^3/4! - 50*x^4/5! - 36*x^5/5! +...

%e ...

%e Coefficients in the initial powers of A(x) begin:

%e [1,(1),(1), 1/2, -1/6, -5/12, -1/20, 49/120, 15/56, -457/1260,...];

%e [1, 2,(3),(3), 5/3, -1/6, -61/60, -17/60, 272/315, 451/630,...];

%e [1, 3, 6,(17/2),(17/2), 21/4, 3/5, -83/40, -187/168, 115/84,...];

%e [1, 4, 10, 18,(73/3),(73/3), 163/10, 131/30, -261/70, -1093/315,...];

%e [1, 5, 15, 65/2, 325/6,(847/12),(847/12), 1205/24, 9551/504,...];

%e [1, 6, 21, 53, 104, 327/2,(4139/20),(4139/20), 6469/42, 7414/105,...];

%e [1, 7, 28, 161/2, 1085/6, 3955/12, 4949/10,(24477/40),(24477/40),...];

%e [1, 8, 36, 116, 878/3, 1810/3, 15569/15, 7509/5,(114760/63),(114760/63), ...]; ...

%e where the coefficients in parenthesis illustrate the property

%e that the coefficients of x^n and x^(n+1) in A(x)^n are equal:

%e [x^n] A(x)^n = [x^(n+1)] A(x)^n = A138013(n)/(n-1)!,

%e where G(x) = e.g.f. of A138013 begins:

%e G(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 146*x^4/4! + 1694*x^5/5! + ...

%e and satisfies: exp(1 - G(x)) = 1 - x*G(x).

%t CoefficientList[1+InverseSeries[Series[x/(1 + Log[1+x]), {x, 0, 20}], x],x] * Range[0, 20]! (* _Vaclav Kotesovec_, Jan 11 2014 *)

%o (PARI) {a(n)=n!*polcoeff(1+serreverse(x/(1+log(1+x+x*O(x^n)))),n)}

%o (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 Cf. A177379, A138013, A038037, A238274.

%K sign

%O 0,3

%A _Paul D. Hanna_, May 14 2010