The Wolfram Language command GeneratingFunction[expr,
n, x] gives the generating function in the variable for the sequence whose th term is expr. Given a sequence of terms, FindGeneratingFunction[a1, a2, ..., x] attempts to find a simple generating function in
whose th
coefficient is .
Given a generating function, the analytic expression for the th term in the corresponding series can be computing using
SeriesCoefficient[expr,
x,
x0, n].
The generating function is sometimes said to "enumerate"
(Hardy 1999, p. 85).
Generating functions giving the first few powers of the nonnegative integers are given in the following table.
series
1
There are many beautiful generating functions for special functions in number theory. A few particularly nice examples are
Generating functions are very useful in combinatorial enumeration problems. For example, the subset sum problem, which asks the number
of ways
to select
out of
given integers such that their sum equals , can be solved using generating functions.
The generating function of of a sequence of numbers is given by the Z-transform
of
in the variable
(Germundsson 2000).
Bender, E. A. and Goldman, J. R. "Enumerative Uses of Generating Functions." Indiana U. Math. J.20, 753-765,
1970/1971.Bergeron, F.; Labelle, G.; and Leroux, P. "Théorie
des espèces er Combinatoire des Structures Arborescentes." Publications
du LACIM. Québec, Montréal, Canada: Univ. Québec Montréal,
1994.Cameron, P. J. "Some Sequences of Integers." Disc.
Math.75, 89-102, 1989.Doubilet, P.; Rota, G.-C.; and Stanley,
R. P. "The Idea of Generating Function." Ch. 3 in Finite
Operator Calculus (Ed. G.-C. Rota). New York: Academic Press, pp. 83-134,
1975.Germundsson, R. "Mathematica Version 4." Mathematica
J.7, 497-524, 2000.Graham, R. L.; Knuth, D. E.;
and Patashnik, O. Concrete
Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley,
1994.Harary, F. and Palmer, E. M. Graphical
Enumeration. New York: Academic Press, 1973.Hardy, G. H.
Ramanujan:
Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York:
Chelsea, p. 85, 1999.Lamdo, S. K. Lectures
on Generating Functions. Providence, RI: Amer. Math. Soc., 2003.Leroux,
P. and Miloudi, B. "Généralisations de la formule d'Otter."
Ann. Sci. Math. Québec16, 53-80, 1992.Riordan,
J. Combinatorial
Identities. New York: Wiley, 1979.Riordan, J. An
Introduction to Combinatorial Analysis. New York: Wiley, 1980.Rosen,
K. H. Discrete
Mathematics and Its Applications, 4th ed. New York: McGraw-Hill, 1998.Sloane,
N. J. A. and Plouffe, S. "Recurrences and Generating Functions."
§2.4 in The
Encyclopedia of Integer Sequences. San Diego, CA: Academic Press, pp. 9-10,
1995.Stanley, R. P. Enumerative
Combinatorics, Vol. 1. Cambridge, England: Cambridge University Press,
p. 63, 1996.Viennot, G. "Une Théorie Combinatoire des
Polynômes Orthogonaux Généraux." Publications du LACIM.
Québec, Montréal, Canada: Univ. Québec Montréal, 1983.Wilf,
H. S. Generatingfunctionology,
2nd ed. New York: Academic Press, 1994.
is a formal power series
.
for the sequence whose 
th term is expr. Given a sequence of terms, FindGeneratingFunction[
a1, a2, ...
, x] attempts to find a simple generating function in
whose 
th
coefficient is 
.
th term in the corresponding series can be computing using
SeriesCoefficient[expr,
x,
x0, n
].
The generating function 
is sometimes said to "enumerate"
(Hardy 1999, p. 85).
is a q-Pochhammer
symbol, and
.
to select 
out of 
given integers such that their sum equals 
, can be solved using generating functions.
of a sequence of numbers 
is given by the Z-transform
of 
in the variable 
(Germundsson 2000).
