# [seqfan] Periodic Signs in Inverse x/(1 + log(1+x)) ?

Paul D Hanna pauldhanna at juno.com
Sun May 16 15:57:25 CEST 2010

```SeqFans,
It's easy to see that a property of exp(x) is:
(1) [x^n] exp(x)^(n+1) = [x^(n+1)] exp(x)^(n+1) for n>=0
where [x^n] G(x) denotes the coefficient of x^n in G(x).

Lets explore a variation of that property.
Suppose a function F(x,m) satisfies:
(2) [x^n] F(x,m)^(n+m) = [x^(n+1)] F(x,m)^(n+m) for n>=0
then we have
(3) F(x,m) = 1/(1 - (m-1)*x/m)^(1/(m-1))
which holds for integer m except at m=1 and m=0.
Examples are
. F(x,3) = 1/sqrt(1 - 2x/3),
. F(x,4) = 1/(1 - 3x/4)^(1/3),
. F(x,-1) = sqrt(1-2x).

Now those cases for which formula (3) does NOT work
are the most interesting to investigate.

As we've seen, for m=1 we have F(x,1) = exp(x);
but it gets a little more tricky at m=0.

I found that
(4) F(x,0) = 1 + Series_Reversion(x/(1 + log(1+x)))
which satisfies:
(5) [x^n] F(x,0)^n = [x^(n+1)] F(x,0)^n for n>=1, and
(6) F(x,0) = 1+x + x*log(F(x,0)).

The coefficients in the initial powers of F(x,0) begin:
[1,(1),(1), 1/2, -1/6, -5/12, -1/20, 49/120, 15/56, ...];
[1, 2, (3),(3), 5/3, -1/6, -61/60, -17/60, 272/315, ...];
[1, 3, 6, (17/2),(17/2), 21/4, 3/5, -83/40, -187/168, ...];
[1, 4, 10, 18, (73/3),(73/3), 163/10, 131/30, -261/70, ...];
[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, ...];
[1, 7, 28, 161/2, 1085/6, 3955/12, 4949/10, (24477/40),(24477/40), ...]; ...
where the above terms in parenthesis illustrate the property that
the coefficients of x^n and x^(n+1) in F(x,0)^n are equal for n>=1.

Aside: this table of coefficients in the powers of F(x,0) allows us to see some interesting related functions.
The diagonal in parenthesis forms A138013 where e.g.f. G(x) satisfies:
(7)  G(x) = 1 - log(1 - x*G(x)).
Also, the main diagonal can be described by d/dx x*H(x) where H(x) satisfies:
(8)  H(x) = 1/(1-x - x*log(H(x))).

I have submitted F(x,0) as A177380 and the H(x) derived from the main diagonal as A177379.

QUESTION.
Do the signs in F(x,0) become periodic after some point?

Looking at the first 561 signs of coefficients in
the series reversion of   x/(1 + log(1+x)),
they seem *almost* of period 47:
++---++--+++--++---++--+++--++---++--+++--+++--
++---++--+++--++---++--+++--++---++--+++--+++--
++---++--+++--++---++--+++--++---++--+++--+++--
++---++--+++--++---++--+++--++---++--+++--+++--
++---++--+++--++---++--+++--++---++--+++--++---
++---++--+++--++---++--+++--++---++--+++--++---
++---++--+++--++---++--+++--++---++--+++--++---
++---++--+++--++---++--+++--++---++--+++--++---
++---++--+++--++---++--+++--++---++--+++--++---
++--+++--+++--++---++--+++--++---++--+++--++---
++--+++--+++--++---++--+++--++---++--+++--++---
++--+++--+++--++---++--+++--++---++--+++--++

Do these signs ever become periodic?
Paul

```