# [seqfan] Re: Functional Inverse

Simon Plouffe simon.plouffe at gmail.com
Wed Apr 21 11:57:48 CEST 2010

```Hello,

there is a way to resolve some cases by using solve
in Maple.

If you have an equation like F(x) = y , if the equation
is solvable you can have directly the solution for x.

It works for simple algebraic expressions.

I used a lot of that in my thesis when we had the first
versions of GFUN.

An example. take A001006 , motzkin numbers .

(direct output from MAPLE) please change the fonts to COURRIER

aa := [1, 1, 2, 4, 9, 21, 51, 127, 323, 835, 2188, 5798, 15511, 41835,
113634, 310572, 853467, 2356779, 6536382, 18199284, 50852019,
142547559,

400763223, 1129760415, 3192727797, 9043402501, 25669818476,
73007772802, 208023278209, 593742784829]

> listtoalgeq(%,x);
Error, (in gfun:-listtoalgeq) invalid unknown function, x
> listtoalgeq(%,f(x));

2     2
[1 + (-1 + x)
f(x) + x  f(x) , ogf]

> op(1,%);

2     2
1 + (-1 + x)
f(x) + x  f(x)

> %=y;

2     2
1 + (-1 + x)
f(x) + x  f(x)  = y

> solve(%,f(x));
2      2   1/2
2      2   1/2
1 - x + (1 - 2 x - 3 x  + 4 x  y)
-1 + x + (1 - 2 x - 3 x  + 4 x  y)
------------------------------------,
- -------------------------------------
2
2
2 x
2 x

In this case it works well.

Is this of any help ?

best regards and bonne journée
simon plouffe

```