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
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).
of 
can given explicitly by
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
,
then
is the double factorial (Borwein and Corless
1999).
is obtained by adding an additional factor of 
,
is what is usually called Stirling's series. It is given by the simple analytic expression
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).
." Amer. Math. Monthly 97, 826-829, 1990.Morse,
P. M. and Feshbach, H. 
." 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.