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Question: Generating functions are useful in studying the number of different types of partitions of an integer n. A partition of a positive integer is a way to write this integer as the sum of positive integers where repetition is allowed and the order of the integers in the sum does not matter. For example, the partitions of 5 (with no restrictions) are 1 + 1 + 1 + 1 + 1, 1 + 1 + 1 + 2, 1 + 1 + 3, 1 + 2 + 2, 1 + 4, 2 + 3, and 5
Show that the coefficient po(n) of xn in the formal power series expansion of 1/((1-x)(1-x3)(1-x5)···) equals the number of partitions of n into odd integers, that is, the number of ways to write n as the sum of odd positive integers, where the order does not matter and repetitions are allowed.
Compare the two sets and describe what you discover.
suppose you have developed a scale that indicates the brightness of the sun. each category in the table is 5 times
The radius of a circular disk is given as 24 cm with a maximum error in measurement of 0.2 cm. Use differentials to estimate the maximum error in the calculated area of the disk
Determine algebraically whether the given function is even, odd, or neither.
Given a line segment of length 1 inch, how do I construct a line segment of squareroot(6), squareroot(7), squareroot(8), etc. Also, given a wire of total length 4236 feet, how do I construct a golden triangle from the wire?
ACCUMULATION YEARS People from their mid-40s to their mid-50s are in the prime investing years. Demographic studies of this type are of particular importance.
The monthly charge for a waste collection service is 660 dollars for 100 kg of waste and 1050 dollars for 165 kg of waste.
Using the definition of the derivative, determine the derivative of the function f(x) = √(2x + 1), at x = 1. Suppose we know that f(x) is differentiable at x = a. For each of the following, determine if the statement is true or false. If true, expl..
Describe the first step to solving the power of a power compared to solving the power of a product of two factors. Give an example of solving a problem containing the power of a power.
Give a counter example and then prove why, i.e., a metric space and a sequence, that disproves the claim: Every Cauchy sequence converges.
Let x be the length of the top of the rectangle (and thus also of the bottom of the rectangle and of the diameter of the circle) and y the length of the side of the rectangle.
A swimming pool is 12 meters long, 6 meters wide, 1 meter deep at the shallow end, and 3 meters deep at the deep end. Water is being pumped into the pool at 1/4 cubic meter per minute, and there is 1 meter of water at the deep end.
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