By Rudenskaya O. G.

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**Sample text**

22) for some constant c. Thus we find that g(s) = I(s) + O(log(s−1 )) as s → 0+ . 20) we see that g(s) ∼ π 2 /(6s) as s → 0+ . 1, we obtain a bound of the form p(n) ≤ exp((1 + o(1))π(2/3)1/2 n1/2 ) as n→∞. 1 yields a lower bound for n! that is only a factor of about n 1/2 away from optimal. That is common. 1 will only be off from the correct value by a polynomial factor of n, and often only by a factor of n 1/2 . 1 can often be improved with some additional knowledge about the fn . For example, if fn+1 ≥ fn for all n ≥ 0, then we have x−n f (x) ≥ fn + fn+1 x + fn+2 x2 + · · · ≥ fn (1 − x)−1 .

There is a substantial difference in the behavior of f 1 (z) and f2 (z) for real z if we let z → −1, so our argument does not completely exclude the possibility of obtaining detailed information about the coefficients of these functions using methods of real variables only. 4) yet now z∈ ✁ . This difference is comparable to what would be obtained by modifying a single coefficient of one generating function. To determine how such slight changes in the behavior of the generating functions affect the behavior of the coefficients we would need to know much more about the functions if we were to use real variable methods.

Then ∞ yn n! 8) k=0 is the exponential generating function of the number of objects of size n, while ∂ h(y) = f (x, y) ∂x ∞ x=1 = n=0 yn n! 9) k=0 is the exponential generating function of the sum of the weights of objects of size n. Therefore the average weight of an object of size n is [y n ]h(y) . 10) The wide applicability and power of generating functions come primarily from the structured way in which most enumeration problems arise. Usually the class of objects to be counted is derived from simpler objects through basic composition rules.