Explanation of the parent function

Defined

generating function, which is a parent function, is an important theory and tool in combinatorial mathematics, especially in counting. There are two kinds of generating function, which are normal type and exponential type, and the common type is more. Formally speaking, the common type generating function is used to solve the combinatorial problem of multiple sets, and the exponential parent function is used to solve the permutation problem of multiple sets. The parent function can also solve the general problem of the recursive series (for example, using a parent function to solve the general formula of the Fibonacci sequence). Normal type female function

Associate any one sequence A0,a1,a2,a3,⋯,an a 0, a 1, a 2, a 3,-A, n a_{0},a_{1},a_{2},a_{3},\cdots, a_{n} in a functional way.

g (x) =a0x0+a1x1+a2x2+a3x3+⋯+anxn g (x) = a 0 x 0 + a 1 x 1 + a 2 x 2 + a 3 x 3 +⋯+ a n x n G (x) =a_{0}x^{0}+a_{1}x^{1}+a _{2}x^{2}+a_{3}x^{3}+\cdots +a_{n}x^{n}

The "G" (x) g (x) g (x) is called the generating function of the sequence.

The common type parent function can solve the combinatorial problem of multiple sets , when you want to solve the combinatorial problem, you can associate the combination number with the coefficient of the parent function.

For example we have a multiset {3⋅a,4⋅b,5⋅c 3⋅a, 4⋅b, 5⋅c 3\cdot a,4\cdot b,5\cdot C}, ask for his 10-combination number
We can write the form of a parent function.

G (x) = (1+x+x2+x3) (1+x+x2+