# [seqfan] A296170 An Unexpected Result

Paul Hanna pauldhanna.math at gmail.com
Thu Dec 7 19:08:26 CET 2017

```SeqFans,
A somewhat trivial property of the exponential function can be stated
as
[x^(n-1)] exp(x)^n = [x^n] exp(x)^n for n>=1.
That is, the coefficients of x^(n-1) and x^n in exp(n*x) are equal.

Well then, what power series A(x) satisfies the similar condition

[x^(n-1)] A(x)^(n^2) = [x^n] A(x)^(n^2) for n>=1.

The answer is described by https://oeis.org/A296170 :
A(x) = 1 + x - x^2/2! - 11*x^3/3! - 239*x^4/4! - 17059*x^5/5! -
2145689*x^6/6! - 412595231*x^7/7! - 111962826751*x^8/8! -
40590007936199*x^9/9! - 18900753214178609*x^10/10! +...

To illustrate [x^(n-1)] A(x)^(n^2) = [x^n] A(x)^(n^2), form a table of
coefficients of x^k in A(x)^(n^2) that begins as
n=1: [(1), (1), -1/2, -11/6, -239/24, -17059/120, -2145689/720, ...];
n=2: [1, (4), (4), -28/3, -196/3, -10472/15, -614264/45, ...];
n=3: [1, 9, (63/2), (63/2), -1701/8, -98217/40, -3168081/80, ...];
n=4: [1, 16, 112, (1232/3), (1232/3), -95648/15, -4835264/45, ...];
n=5: [1, 25, 575/2, 11725/6, (190225/24), (190225/24), ...];
n=6: [1, 36, 612, 6444, 45684, (1043784/5), (1043784/5), ...];
n=7: [1, 49, 2303/2, 102949/6, 4313617/24, 164086349/120, (5086480231/720),
(5086480231/720), ...];
...
then we can see that the diagonals (indicated by parenthesis) are equal.

But here is the unexpected part.

The logarithm of the e.g.f. A(x) is an integer series (at least for the
initial 200 terms):

log(A(x)) = x - x^2 - x^3 - 9*x^4 - 134*x^5 - 2852*x^6 - 79096*x^7 -
2699480*x^8 - 109201844*x^9 - 5100872244*x^10 - 269903909820*x^11 -
15944040740604*x^12 +...

(See https://oeis.org/A296171 for more terms)

It would be nice if someone would verify that the terms remain integral for
more terms (using a different program than my PARI code).

If this holds, then why would the logarithm of A(x) be an integer series?
Is there something significant behind this?
Paul
```