gfun[listtoalgeq] - find an algebraic equation satisfied by a generating function

gfun[seriestoalgeq] - find an algebraic equation satisfied by a series

listtoalgeq(l, y(x), <[typelist]>)

seriestoalgeq(s, y(x), <[typelist]>)

l - list

s - series

y,z - name of the unknown function and the generic variable

[typelist] - (optional) list of generating function types

The procedures listtoalgeq and seriestoalgeq compute a polynomial equation in y and x satisfied by the generating function y(x) of the expressions in l or s, this generating function being of one of the types specified by typelist, for example, ordinary (ogf) or exponential (egf). For a full list of available choices see gftypes).
If typelist contains more than one element, these types are tried in order.
If typelist is not provided, the default optionsgf, initially ['ogf'], is used. The output is a tagged list whose first element is the polynomial in y(x) and x that was found, and whose second element is the type to which it corresponds.
The search is controlled by the options minordereqn, maxordereqn, maxdegcoeffs, maxtotaldegree, checkterms, mode, and modulo. With maxtotaldegree=T, the search restricts the total degree in x and y of the algebraic polynomial to at most T; thus the coefficient of y^i starts with degree bound T-i. For more information on these options, see gfun[parameters].
If sufficiently many terms are given and no solution is found, then no algebraic equation was found within the requested order, degree, coefficient-size, and validation bounds.

> with(gfun):
l:=[1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, 58786]:
listtoalgeq(l,y(x));

\[1-y \! \left(x \right)+x y \! \left(x \right)^{2}\]

> s:=series((1-sqrt(1-4*x)),x,9);

\[2 x +2 x^{2}+4 x^{3}+10 x^{4}+28 x^{5}+84 x^{6}+264 x^{7}+858 x^{8}+\mathrm{O}\! \left(x^{9}\right)\]

> seriestoalgeq(s,y(x));

\[4 x -2 y \! \left(x \right)+y \! \left(x \right)^{2}\]