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1 we find that π 1/2 k0n (k0 − 1)n n1/2 2n (k0 − 1)! an ∼ as n→∞. an as n→∞. 1. For example, if fn is the number of connected graphs on n labeled vertices which have some property, and Fn is the number of graphs on n labeled vertices each of whose connected components has that property, then (cf. ) ∞ Fn 1+ n=1 xn = exp n! ∞ fn n=1 xn n! 2. 11) ∞ bn x b(x) = n = F (x, a(x)) , D(x) = Fy (x, a(x)) , n=0 where Fy (x, y) is the partial derivative of F (x, y) with respect to y. 13) (iii) for every δ > 0 there are M (δ) and K(δ) such that for n ≥ M (δ) and h + k > r + 1, |fhk an−h−k+1 | ≤ K(δ)δ h+k |an−r | .

Therefore Eq. 66) = n−1 nn−1 /(n − 1)! = nn−1 /n! , which shows that tn , the number of rooted labeled trees on n nodes, is n n−1 . Proof of a form of the Lagrange-B¨ urmann theorem is given in Chapter ?. Extensive discussion, proofs, and references are contained in [81, 173, 205, 375]. Some additional recent references are [159, 208]. There exist generalizations of the Lagrange-B¨ urmann formula to several variables [173, 169, 208]. The Lagrange-B¨ urmann formula, as stated above, is valid for general formal power series.

65), say, in full generality, it suffices to prove it for any n. 65) can be applied. Thus combinatorial proofs of the Lagrange-B¨ urmann formula do not offer greater generality than analytic ones. While the analytic vs. combinatorial distinction in the proofs of the Lagrange-B¨ urmann formula does not matter, it is possible to use analyticity of the functions f (z) and g(z) to obtain useful information. 6 above was atypical in that a simple explicit formula 44 was derived. 64) is not explicit enough to make clear its asymptotic behavior.

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