Generating Functions
Generating a Sequence Via a Function
A generating function, g(x), can be expressed as a power series polynomial. The coefficients of such polynomials make up the generating sequence.
g(x) = Σ( c_k * x^k, k=0 to ∞) = c_0 + c_1 * x + c_2 * x^2 + c_3 * x^3 + ....
The generated sequence is {c_0, c_1, c_2, c_3, ... }.
A simple approach is find the Maclaurin Series (Taylor Series about the point 0) of g(x).
Example:
g(x) = 2 ÷ (1 - 3*x)
The Maclaurin Series of g(x) is:
2 + 6*x + 18*x^2 + 54*x^3 + 162*x^4 + 486*x^5 + ...
with the sequence of { 2, 6, 18, 54, 162, 486 ... }
Some Simple Generating Functions
1 ÷ (1 - x) = 1 + x + x^2 + x^3 + x^4 + ....
b ÷ (1 - x) = b + b + b^2 + b^3 + b^4 + ....
1 ÷ (1 - a*x) = 1 + a*x + a^2*x^2 + a^3*x^3 + a^4*x^4 + ....
1 ÷ (1 - x)^2 = 1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + ....
1 ÷ (1 + x) = 1 - x + x^2 - x^3 + x^4 - x^5 + ....
(as pictures the size of 3" x 5" index cards: let me know if you want more posts like this - Eddie)
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