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23 11

Article 10.6.6

Journal of Integer Sequences, Vol. 13 (2010), 2

3 6 1 47

## !

### nemesgery@gmail.com

Abstract

Applying a theorem of Howard to a formula recently proved by Brassesco and M´endez, we derive new simple explicit formulas for the coefficients of the asymptotic expansion of the sequence of factorials.

### Introduction

It is well known that the factorial of a positive integern has the asymptotic expansion

n!∼nne−n√2πnX

k≥0

ak

nk, (1)

known as Stirling’s formula (see, e.g., [1, 3, 4]). The coefficients ak in this series are usually

called the Stirling coefficients [1, 6] (Sloane’s A001163 and A001164) and can be computed from the sequencebk defined by the recurrence relation

bk=

1

k+ 1 bk−1−

k−1

X

j=2

jbjbk−j+1

!

, b0 =b1 = 1, (2)

(2)

representation for the Stirling coefficients”. However there is a closed-form expression that involves combinatorial quantities due to Comtet [5]:

ak =

whered3(p, q) is the number of permutations ofpwithq permutation cycles all of which are ≥3 (Sloane’s A050211). Brassesco and M´endez [7] proved in a recent paper that

ak=

whereS3(p, q) denotes the 3-associated Stirling numbers of the second kind (Sloane’sA059022). We show that the Stirling coefficientsak can be expressed in terms of the conventional

Stir-ling numbers of the second kind (Sloane’sA008277). A corollary of this result is an explicit, exact expression for the Stirling coefficients.

### The formulas for coefficients

One of our main results is the following:

Theorem 1. The Stirling coefficients have a representation of the form

ak=

where S(p, q) denotes the Stirling numbers of the second kind.

From the explicit formula

S(p, q) = 1

we immediately obtain our second main result.

Corollary 2. The Stirling coefficients have an exact representation of the form

ak =

n in the variable z are

defined by the exponential generating function

(3)

Forr1, the exponential Bell polynomials Bn,i(0, . . . ,0, ar, ar+1, . . .) in an infinite number of variables ar, ar+1, . . . can be defined by

(F (x))i =i!X

n≥0

Bn,i(0, . . . ,0, ar, ar+1, . . .)

xn

n!. (8)

The following theorem is due to Howard [2].

Theorem 3. If Fn(z) is defined by (7) and Bn,i is defined by (8), then

F(z)

n =

n

X

i=0 (1)i

z+i1

i

z+n n−i

r!

ar

i

n!i!

(n+ri)!Bn+ri,i(0, . . . ,0, ar, ar+1, . . .). (9) Now we prove Theorem 1.

Proof of Theorem 1. Brassesco and M´endez showed that if

G(x) = 2e

x

−x1

x2 = 2

X

j≥0

xj

(j + 2)!, (10)

then

ak=

1 2kk!

2kG−2k2+1

(0), (11)

where ∂kf denotes the kth derivative of a function f. Define the polynomials G(z)

n in the

variable z by the following exponential generating function:

1 2

x2

exx1

z

=X

j≥0

G(jz)x

j

j!. (12)

Inserting z= 2k+1

2 into this expression gives

X

j≥0

G( 2k+1

2 )

j

xj

j! =

1 2

x2

exx1

2

k+1 2

=

2e

x

−x−1

x2

−2k2+1

=G−2k2+1

(x). (13)

On the other hand we have by series expansion

G−2k2+1

(x) =X

j≥0

∂jG−2k2+1 (0)x

j

j!. (14)

Equating the coefficients in (13) and (14) gives

∂jG−2k2+1

(0) =G( 2k+1

2 )

j =G

(k+1 2)

j .

Now by comparing this with (11) yields

ak =

1 2kk!G

(k+1 2)

(4)

Putting r = 2 an ar = ar+1 = . . . = 1 into the formal power series F (x) = Pj≥rajx j/j!

gives F (x) = ex

−x1. And therefore the generated potential polynomials are

According to Howard’s theorem we find

G(z)

of the Stirling numbers of the second kind:

i!X Thus we obtain

(5)

### Acknowledgments

I am grateful to Lajos L´aszl´o, who drew my attention to the paper of Brassesco and M´endez. I also would like to thank the referees for their valuable remarks that help to improve the initial version of this paper.

### References

[1] C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers, Mcgraw–Hill Book Company, 1978, 218.

[2] F. T. Howard, A theorem relating potential and Bell polynomials, Discrete Math. 39

(1982), 129–143.

[3] G. Marsaglia and J. C. Marsaglia, A new derivation of Stirling’s approximation to n!,

Amer. Math. Monthly 97 (1990), 826–829.

[4] J. M. Borwein and R. M. Corless, Emerging tools for experimental mathematics, Amer. Math. Monthly 106 (1999), 899–909.

[5] L. Comtet, Advanced Combinatorics: The Art of Finite and Infinite Expansions, D. Reidel Publishing Company, 1974, p. 267.

[6] R. B. Paris and D. Kaminski, Asymptotics and Mellin–Barnes Integrals, Cambridge University Press, 2001, p. 32.

[7] S. Brassesco and M. A. M´endez, The asymptotic expansion forn! and Lagrange inversion formula, http://arxiv.org/abs/1002.3894.

2010 Mathematics Subject Classification: Primary 11B65; Secondary 11B73, 41A60.

Keywords: asymptotic expansions, factorial, Stirling coefficients, Stirling’s formula, Stirling numbers.

(Concerned with sequences A001163, A001164,A008277, A050211, and A059022.)

Received March 15 2010; revised version received June 17 2010. Published in Journal of Integer Sequences, June 21 2010.

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