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gfun

 gftypes
 describe available types of generating functions

Description

 • A generating function is an analytic encoding of numerical data. It is a formal power series which can be manipulated algebraically in ways which parallel the manipulation of the (often combinatorial) objects they represent. The gfun package recognizes several different ways to represent the information in a list l.
 • The following types of generating functions are accepted by the gfun package.
 'ogf'
 If type is $'\mathrm{ogf}'$ (ordinary generating function), then the coefficients are the elements of l. For example, the $\mathrm{ogf}$ which corresponds to the list, [1, 1, 2, 3, 5, 8], is $8{x}^{5}+5{x}^{4}+3{x}^{3}+2{x}^{2}+x+1$.
 'egf'
 If type is $'\mathrm{egf}'$ (exponential generating function), then the ith coefficient is $\frac{\mathrm{op}\left(i,l\right)}{i!}$. For example, the $\mathrm{egf}$ which corresponds to  to the list, [1, 1, 2, 3, 5, 8], is $1+x+\frac{2{x}^{2}}{2!}+\frac{3{x}^{3}}{3!}+\frac{5{x}^{4}}{4!}+\frac{8{x}^{5}}{5!}$.
 'revogf'
 If type is $'\mathrm{revogf}'$, then the series is the reciprocal of the ordinary generating function.
 'revegf'
 If type is $'\mathrm{revegf}'$, then the series is the reciprocal of the exponential generating function.
 'lgdogf'
 If type is $'\mathrm{lgdogf}'$, then the series is the logarithmic derivative of the ordinary generating function.
 'lgdegf'
 If type is $'\mathrm{lgdegf}'$, then the series is the logarithmic derivative of the exponential generating function.
 'Laplace'
 If type is $'\mathrm{Laplace}'$, then the ith coefficient is $\mathrm{op}\left(i,l\right)i!$.
 • You can define types by creating a procedure gfun[listtoseries/mytypeofgf], which accepts a list and a variable as input, and yields a series in this variable. This series must be of type taylor. In particular, it cannot have negative exponents.