Determine the generating function for the number hn of bags of fruit of apples, oranges, bananas

Determine the generating function for the number hn of bags...

Key Concepts

Generating Functions

Generating functions are formal power series in which the coefficient of each power of x represents the number of ways an event or configuration of a particular size can occur. They are powerful tools in combinatorics for encoding sequences and handling counting problems, particularly when different parts of a combinatorial object contribute independently to the overall count.

Encoding Parity Constraints

Parity constraints, such as requiring an even number of a certain object, can be encoded in generating functions by modifying the summation so that only even exponents appear. This is typically achieved by summing the series x^(2k) for non-negative integers k, which compactly represents all possibilities that satisfy the even-number condition.

Handling Bounded Quantities

When an element can appear only a limited number of times (for example, at most two occurrences), its generating function is a finite polynomial where the exponents indicate the possible counts. This finite sum directly captures the constraint by excluding higher exponents that would represent forbidden counts.

Modeling Multiplicity Constraints

Constraints that require the count of an item to be a multiple of a fixed number are modeled in generating functions by summing only those terms corresponding to the allowed multiples. For instance, a requirement that the count be a multiple of three is represented by a series whose exponents are all multiples of three.

Coefficient Extraction

After forming the generating function that models all imposed constraints, the number of configurations corresponding to a fixed size is obtained by extracting the coefficient of x^n. This process of coefficient extraction connects the algebraic representation back to the combinatorial interpretation and yields the final counting formula.

Transcript

00:01 We are given a sequence and we are asked to find a close form for the generating function for this sequence.

00:08 In part a, the sequence is defined as a sub -n is equal to 5.

00:21 For all n equal to 0, 1, and so on.

00:28 We have the generating function for this sequence, g of x, is equal to the sum from n equals 0 to infinity.

00:39 Of a sub n x to the n, which in this case is going to be the sum from n equals 0 to infinity of 5 x to the n.

00:49 And we can pull a 5 out, so this is the sum from n to equal 0 to infinity of x to the n.

00:58 And recall that we have by a theorem, this is 5 times 1 over 1 minus x.

01:05 So this is equal to 5 over 1 minus x.

01:14 In part b, the sequence is defined as a.

01:17 X of n is equal to 3 to the n for all n.

01:34 We have the generating function for this sequence, g of x is equal to the sum from n equals 0 to infinity of a sub n x to the n by definition.

01:46 And this is equal to plugging in a sub n, sum from n equals 0 to infinity of 3 to the n, x to the n.

01:55 And we can write this using power laws as the sum from n equals 0 to infinity of 3x, to the n and we have by a theorem this is equal to 1 over 1 minus 3x in part c we're given the sequence a sub n where a sub n equals 2 for n equal to 3 4 and 5 and a 0 equals a 0 equals a sub 1 equals a sub 2 equals 0 the generating function for this sequence g of x is by definition the sum from a n equals 0 to infinity of a sub n x to the n which is equal to substituting in the sequence the sum from n equals 3 to infinity of 2 x to the n which is equal to 2 times the sum from n equals 3 to infinity x to the n which is equal to 2 times the sum from n equals 3 to infinity x to the n which is and then we'll say, make the substitution, t equals n minus 3.

03:55 This is now the sum from, well, if n is equal to 3, then t starts at 0.

04:02 So t equals 0 to infinity of x to the, and then t is equal to n minus 3.

04:12 So n is equal to t plus 3.

04:18 And this is equal to 2 times x cubed times the sum from, and now changing back, using n equals t we have the sum from n equals 0 to infinity of x to the n and this is equal to 2x cubed in by a theorem times 1 over 1 minus x which is equal to 2x cubed over 1 minus x in part d we're given the sequence a sub n equals 2 n plus 3 for all n the generating function for this sequence g of x is by definition this sum from n equals 0 to infinity of a sub n x to the n and by substitution this is equal to the sum from n equals 0 to infinity of 2 n plus 3 x to the n and we can rewrite this sum as two times the sum from n equals 0 to infinity of n x to the n plus the sum from n equals 0 or plus 3 times the sum from n equals 0 to infinity of x to the n.

06:06 And we have by a table from the book, this is equal to 2 times 1 over 1 minus x squared, plus 3 times 1 over 1 minus x by a theorem...

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