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

1, 1, 2, 3, -4, -50, -36, 2058, 10800, -131616, -1975680, 7741800, 417480480, 1307617584, -101626746144, -1284067345680, 25419094122240, 791333924647680, -3900043588999680, -472446912421801728, -3183064994777932800

Offset

0,3

Comments

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

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..420

Formula

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

...

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

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

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

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

...

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

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

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

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

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

Example

E.g.f: A(x) = 1 + x + 2*x^2/2! + 3*x^3/3! - 4*x^4/4! - 50*x^5/5! +...
log(A(x)) = 2*x/2! + 3*x^2/3! - 4*x^3/4! - 50*x^4/5! - 36*x^5/5! +...
...
Coefficients in the initial powers of A(x) begin:
[1,(1),(1), 1/2, -1/6, -5/12, -1/20, 49/120, 15/56, -457/1260,...];
[1, 2,(3),(3), 5/3, -1/6, -61/60, -17/60, 272/315, 451/630,...];
[1, 3, 6,(17/2),(17/2), 21/4, 3/5, -83/40, -187/168, 115/84,...];
[1, 4, 10, 18,(73/3),(73/3), 163/10, 131/30, -261/70, -1093/315,...];
[1, 5, 15, 65/2, 325/6,(847/12),(847/12), 1205/24, 9551/504,...];
[1, 6, 21, 53, 104, 327/2,(4139/20),(4139/20), 6469/42, 7414/105,...];
[1, 7, 28, 161/2, 1085/6, 3955/12, 4949/10,(24477/40),(24477/40),...];
[1, 8, 36, 116, 878/3, 1810/3, 15569/15, 7509/5,(114760/63),(114760/63), ...]; ...
where the coefficients in parenthesis illustrate the property
that the coefficients of x^n and x^(n+1) in A(x)^n are equal:
[x^n] A(x)^n = [x^(n+1)] A(x)^n = A138013(n)/(n-1)!,
where G(x) = e.g.f. of A138013 begins:
G(x) = 1 + x + 3*x^2/2! + 17*x^3/3! + 146*x^4/4! + 1694*x^5/5! + ...
and satisfies: exp(1 - G(x)) = 1 - x*G(x).

Mathematica

CoefficientList[1+InverseSeries[Series[x/(1 + Log[1+x]), {x, 0, 20}], x], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 11 2014 *)

Prog

(PARI) {a(n)=n!*polcoeff(1+serreverse(x/(1+log(1+x+x*O(x^n)))), n)}
(PARI) {a(n)=local(A=1+x); for(i=1, n, A=1+x+x*log(A+O(x^n))); n!*polcoeff(A, n)}

Crossrefs

Cf. A177379, A138013, A038037, A238274.

Sequence in context: A235495 A257482 A023167 * A282034 A096782 A098811

Adjacent sequences: A177377 A177378 A177379 * A177381 A177382 A177383

Keyword

sign

Author

Paul D. Hanna, May 14 2010

Status

approved