This function finds the rational generating function of the sequence defined by Fn.

The expression Fn must be a valid closed form expression for a sequence with a rational generating function. The command type(Fn, 'ratseq'(n)) will determine if Fn is a valid expression.

The sequence is assumed to be defined by Fn for all $0\le n$. The optional arguments can be used to specify other information about the sequence.

An optional argument of the form $n=c$, where c is an integer value, defines the first nonzero term in the sequence to be at index c.

An optional argument of the form $\[\mathrm{i1}=\mathrm{v1},\mathrm{i2}=\mathrm{v2},...\]$, where $\mathrm{i2},\mathrm{i2},\mathrm{...}$ are integer indices and $\mathrm{v1},\mathrm{v2},\mathrm{...}$ are values of the sequence at those indices.

The command with(genfunc,rgf_encode) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{genfunc}\right)\:$

$\mathrm{rgf\_encode}\left(2^n\,n\,z\right)$

$\frac{1}{1-2z}$

$\mathrm{rgf\_encode}\left(2^k\,k\,y\,k=2\right)$

$\frac{4y^2}{1-2y}$

$\mathrm{rgf\_encode}\left(2^n\,n\,z\,\left[0=2\,1=0\right]\right)$

$\frac{1}{1-2z}+1-2z$