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).