Re: generating function solution to poly

[Kurt Foster <drsardonicus@earthlink.net>]

On Jul 21, 2018, at 2:38 AM, Kevin Ryde wrote:

> I have a generating function (and more terms too)
>
>    g = x^2 + x^3 + x^4 + 3*x^5 + 6*x^6 + 12*x^7 + 29*x^8 + 67*x^9 +  
> O(x^10);
>
> which satisfies a cubic
>
>    (1+x)*g^3 - 2*g^2 + (1-x+2*x^2)*g - x^2 == 0
>
> Is there an easy or good way to have gp solve that for series g?
>
> I know how to work upwards to get g term by term (after deciding  
> lowest
> should be 0), but maybe gp already has it.  I wondered only for  
> interest
> or generality though, since this one comes from recurrences which are
> easy to calculate.  I attempted serreverse() without joy (change
> variables to solve in x, but I may have confused myself).

On Jul 22, 2018, at 2:15 AM, Peter Pein wrote:

> Hi Kevin,
>
>
> I'm just curious:  are you sure the coefficients in g are correct?
>
> starting Bill Allombert's suggestion to iterate via Newton but  
> starting
> with just X^2+O(X^3)  gives another sequence for g and a difference  
> for
> coeffs of x^7 and higher powers
>
>
> ? P=substvec((1+x)*g^3-2*g^2+(1+x+2*x^2)*g-x^2,[x,g],[X,y])
> %1 = (X + 1)*y^3 - 2*y^2 + (2*X^2 + X + 1)*y - X^2

Different problem, different solution.  Compare coefficients of y,  1-x 
+2*x^2 and 2*X^2 + X + 1.  Never mind the change of variable name or  
order of terms.  A minus sign has ben changed to a plus sign.

Take it from a holder of a Black Belt of transfinite degree in making  
typos -- copy-paste is the way to go in transcribing algebraic  
expressions.