Stirling's Series
The asymptotic series for the gamma function is given by
 |
(1)
|
The coefficient 
of 
can given
explicitly by
 |
(2)
|
where 
is the number of permutations
of 
with 
permutation
cycles all of which are 
(Comtet 1974,
p. 267). Another formula for the 
s is given by
the recurrence relation
 |
(3)
|
with 
, then
 |
(4)
|
where 
is the double
factorial (Borwein and Corless 1999).
The series for 
is obtained by adding an additional
factor of 
,
 |  |  |
(5)
|
 |  |  |
(6)
|
The expansion of 
is what is usually called Stirling's
series. It is given by the simple analytic expression
 |  |  |
(7)
|
 |  |  |
(8)
|
(OEIS A046968 and A046969), where 
is a Bernoulli
number. Interestingly, while the numerators in this expansion are the same as
those of 
for the first several hundred
terms, they differ at 
, 1185, 1240, 1269, 1376, 1906, 1910,
... (OEIS A090495), with the corresponding
ratios being 37, 103, 37, 59, 131, 37, 67, ... (OEIS A090496).
Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 257, 1972.
Arfken, G. "Stirling's Series." §10.3 in Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 555-559, 1985.
Borwein, J. M. and Corless, R. M. "Emerging Tools for Experimental Mathematics." Amer. Math. Monthly 106, 899-909, 1999.
Comtet, L. Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, p. 267, 1974.
Conway, J. H. and Guy, R. K. "Stirling's Formula." In The Book of Numbers. New York: Springer-Verlag, pp. 260-261, 1996.
Havil, J. Gamma: Exploring Euler's Constant. Princeton, NJ: Princeton University Press, pp. 86-88, 2003.
Marsaglia, G. and Marsaglia, J. C. "A New Derivation of Stirling's Approximation to 
." Amer. Math. Monthly 97,
826-829, 1990.
Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, p. 443, 1953.
Sloane, N. J. A. Sequences A001163/M5400, A001164/M4878, A046968, A046969, A090495, and A090496 in "The On-Line Encyclopedia of Integer Sequences."
Uhler, H. S. "The Coefficients of Stirling's Series for 
."
Proc. Nat. Acad. Sci. U.S.A. 28, 59-62, 1942.
Wrench, J. W. Jr. "Concerning Two Series for the Gamma Function." Math. Comput. 22, 617-626, 1968.
Weisstein, Eric W. "Stirling's Series." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/StirlingsSeries.html
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series of n! at n
= inf to order 10