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In this section we give formulas for generating functions enumerating the sequence {f an + b} given an ordinary generating function F(z), where a ≥ 2, 0 ≤ b < a, and a and b are integers (see the main article on transformations).
One possible generating function for such partitions, taking k fixed and n variable, is = =. More generally, if T is a set of positive integers then the number of partitions of n, all of whose parts belong to T, has generating function
The multiplicative inverse of its generating function is the Euler function; by Euler's pentagonal number theorem this function is an alternating sum of pentagonal number powers of its argument. Srinivasa Ramanujan first discovered that the partition function has nontrivial patterns in modular arithmetic, now known as Ramanujan's congruences.
If X is a discrete random variable taking values x in the non-negative integers {0,1, ...}, then the probability generating function of X is defined as [1] = = = (),where is the probability mass function of .
7 Generating function. 8 Generalizations. 9 Padovan L-system. 10 Cuboid spiral. ... In number theory, the Padovan sequence is the sequence of integers P(n) defined [1
Formal power series are widely used in combinatorics for representing sequences of integers as generating functions. In this context, a recurrence relation between the elements of a sequence may often be interpreted as a differential equation that the generating function satisfies.
(The article on unrestricted partition functions discusses this type of generating function.) For example, the coefficient of x 5 is +1 because there are two ways to split 5 into an even number of distinct parts (4 + 1 and 3 + 2), but only one way to do so for an odd number of distinct parts (the one-part partition 5).
For integers a and b, ... is a generating function for the sequence c 1 (n), c 2 (n), ... where n is kept constant. There is also the double Dirichlet series