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\vskip 1cm{\LARGE\bf On the Coefficients of the Asymptotic \\
\vskip .1in
Expansion of $n!$
}
\vskip 1cm
\large
Gerg\H{o} Nemes \\
Lor\'and E\"otv\"os University \\
H-1117 Budapest \\
P\'azm\'any P\'eter s\'et\'any 1/C \\
Hungary \\
\href{mailto:nemesgery@gmail.com}{\tt nemesgery@gmail.com} \\
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\begin{abstract}
Applying a theorem of Howard to a formula recently
proved by Brassesco and M\'{e}ndez, we derive new simple explicit
formulas for the coefficients of the asymptotic expansion of the
sequence of factorials.
\end{abstract}


\theoremstyle{plain}
\newtheorem{theorem}{Theorem}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}

\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{conjecture}[theorem]{Conjecture}

\theoremstyle{remark}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

It is well known that the factorial of a positive integer $n$ has the asymptotic expansion
\begin{equation}\label{eq1}
n! \sim n^n e^{ - n} \sqrt {2\pi n} \sum\limits_{k \ge 0} {\frac{{a_k }}{{n^k }}} ,
\end{equation}
known as Stirling's formula (see, e.g., \cite{ref1,ref3,ref4}). The coefficients $a_k$ in this series are usually called the Stirling coefficients \cite{ref1,ref6} (Sloane's \seqnum{A001163} and \seqnum{A001164}) and can be computed from the sequence $b_k$ defined by the recurrence relation
\begin{equation}
b_k = \frac{1}{{k + 1}}\left( {b_{k - 1}  - \sum\limits_{j = 2}^{k - 1} {jb_j b_{k - j + 1} } } \right),\;b_0 = b_1 = 1,
\end{equation}
since $a_k  = \left( {2k + 1} \right)!!b_{2k + 1}$ \cite{ref3,ref4}. Here $\left( {2k + 1} \right)!! = \left( {2k + 1} \right) \cdot \left( {2k - 1} \right) \cdots 5 \cdot 3 \cdot 1$ is the double factorial. It was pointed out by Paris and Kaminski \cite{ref6} that ``There is no known closed-form representation for the Stirling coefficients''. However there is a closed-form expression that involves combinatorial quantities due to Comtet \cite{ref5}:
\begin{equation}
a_k  = \sum\limits_{j = 0}^{2k} {\left( { - 1} \right)^j \frac{{d_3 \left( {2k + 2j,j} \right)}}{{2^{k + j} \left( {k + j} \right)!}}},
\end{equation}
where $d_3\left(p,q\right)$ is the number of permutations of $p$ with $q$ permutation cycles all of which are $\geq 3$ (Sloane's \seqnum{A050211}). Brassesco and M\'{e}ndez \cite{ref7} proved in a recent paper that
\begin{equation}
a_k  = \sum\limits_{j = 0}^{2k} {\left( { - 1} \right)^j \frac{{S_3 \left( {2k + 2j,j} \right)}}{{2^{k + j} \left( {k + j} \right)!}}},
\end{equation}
where $S_3\left(p,q\right)$ denotes the $3$-associated Stirling numbers of the second kind (Sloane's \seqnum{A059022}). We show that the Stirling coefficients $a_k$ can be expressed in terms of the conventional Stirling numbers of the second kind (Sloane's \seqnum{A008277}). A corollary of this result is an explicit, exact expression for the Stirling coefficients.

\section{The formulas for coefficients}

One of our main results is the following:
\begin{theorem}\label{thm1} The Stirling coefficients have a representation of the form
\begin{equation}
a_k  = \frac{\left(2k\right)!}{2^k k!}\sum\limits_{i = 0}^{2k} {\binom{k + i - 1/2}{i}\binom{3k + 1/2}{2k - i}2^{i}\sum\limits_{j = 0}^i {\binom{i}{j}\left( { - 1} \right)^j j!} \frac{{S\left( {2k + i + j,j} \right)}}{{\left( {2k + i + j} \right)!}}},
\end{equation}
where $S\left(p,q\right)$ denotes the Stirling numbers of the second kind.
\end{theorem}
From the explicit formula
\[
S\left( {p,q} \right) = \frac{1}{{q!}}\sum\limits_{l = 0}^q {\left( { - 1} \right)^l \binom{q}{l}\left( {q - l} \right)^p } ,
\]
we immediately obtain our second main result.
\begin{corollary} The Stirling coefficients have an exact representation of the form
\begin{equation}
a_k  = \frac{\left(2k\right)!}{2^k k!}\sum\limits_{i = 0}^{2k} {\binom{k + i - 1/2}{i}\binom{3k + 1/2}{2k - i}2^{i}\sum\limits_{j = 0}^i {\binom{i}{j}\frac{\left( { - 1} \right)^j}{\left( {2k + i + j} \right)!}}\sum\limits_{l = 0}^j {\left( { - 1} \right)^l \binom{j}{l}\left( {j - l} \right)^{2k + i + j} }} .
\end{equation}
\end{corollary}

To prove Theorem \ref{thm1} we need some concepts. Let $r \geq 0$ and $a_r \neq 0$, let $F\left( x \right) = \sum\nolimits_{j \ge r} {a_j x^j / j!}$ be a formal power series. The potential polynomials $F^{\left(z\right)}_n$ in the variable $z$ are defined by the exponential generating function
\begin{equation}\label{pp1}
\left( {\frac{{a_r x^r / r!}}{{F\left( x \right)}}} \right)^z  = \sum\limits_{n \ge 0} {F_n^{\left( z \right)} \frac{{x^n }}{{n!}}} .
\end{equation}
For $r \geq 1$, the exponential Bell polynomials $B_{n,i}\left(0,\ldots,0,a_r,a_{r+1},\ldots\right)$ in an infinite number of variables $a_r, a_{r+1},\ldots$ can be defined by
\begin{equation}\label{pp2}
\left( {F\left( x \right)} \right)^i  = i!\sum\limits_{n \ge 0} {B_{n,i} \left( {0, \ldots ,0,a_r ,a_{r + 1} , \ldots } \right)\frac{{x^n }}{{n!}}} .
\end{equation}
The following theorem is due to Howard \cite{ref2}.
\begin{theorem} If $F^{\left( z \right)}_n$ is defined by \eqref{pp1} and $B_{n,i}$ is defined by \eqref{pp2}, then
\begin{equation}
F_n^{\left( z \right)}  = \sum\limits_{i = 0}^n {\left( { - 1} \right)^i \binom{z + i - 1}{i}\binom{z + n}{n - i}\left( {\frac{{r!}}{{a_r }}} \right)^{i}\frac{{n!i!}}{{\left( {n + ri} \right)!}}B_{n + ri,i} \left( {0, \ldots ,0,a_r ,a_{r + 1} , \ldots } \right)} .
\end{equation}
\end{theorem}

Now we prove Theorem \ref{thm1}.

\begin{proof}[Proof of Theorem \ref{thm1}]
Brassesco and M\'{e}ndez showed that if
\begin{equation}
G\left( x \right) = 2\frac{{e^x  - x - 1}}{{x^2 }} = 2\sum\limits_{j \ge 0} {\frac{{x^j }}{{\left( {j + 2} \right)!}}},
\end{equation}
then
\begin{equation}\label{pf1}
a_k  = \frac{1}{{2^k k!}}\partial ^{2k} \left( {G^{ - \frac{{2k + 1}}{2}} } \right)\left( 0 \right),
\end{equation}
where $\partial ^{k}f$ denotes the $k$th derivative of a function $f$. Define the polynomials $G^{\left(z\right)}_n$ in the variable $z$ by the following exponential generating function:
\begin{equation}
\left( {\frac{1}{2}\frac{{x^2 }}{{e^x  - x - 1}}} \right)^z  = \sum\limits_{j \ge 0} {G_j^{\left( z \right)} \frac{{x^j }}{{j!}}} .
\end{equation}
Inserting $z=\frac{2k+1}{2}$ into this expression gives
\begin{equation}\label{pf2}
\sum\limits_{j \ge 0} {G_j^{\left( {\frac{{2k + 1}}{2}} \right)} \frac{{x^j }}{{j!}}}  = \left( {\frac{1}{2}\frac{{x^2 }}{{e^x  - x - 1}}} \right)^{\frac{{2k + 1}}{2}}  = \left( {2\frac{{e^x  - x - 1}}{{x^2 }}} \right)^{ - \frac{{2k + 1}}{2}}  = G^{ - \frac{{2k + 1}}{2}} \left( x \right).
\end{equation}
On the other hand we have by series expansion
\begin{equation}\label{pf3}
G^{ - \frac{{2k + 1}}{2}} \left( x \right) = \sum\limits_{j \ge 0} {\partial ^j \left( {G^{ - \frac{{2k + 1}}{2}} } \right)\left( 0 \right)\frac{{x^j}}{{j!}}} .
\end{equation}
Equating the coefficients in \eqref{pf2} and \eqref{pf3} gives
\[
\partial ^j \left( {G^{ - \frac{{2k + 1}}{2}} } \right)\left( 0 \right) = G_j^{\left( {\frac{{2k + 1}}{2}} \right)}  = G_j^{\left( {k + \frac{1}{2}} \right)} .
\]
Now by comparing this with \eqref{pf1} yields
\begin{equation}\label{pf4}
a_k  = \frac{1}{{2^k k!}}G_{2k}^{\left( {k + \frac{1}{2}} \right)} .
\end{equation}
Putting $r=2$ an $a_r = a_{r+1} = \ldots = 1$ into the formal power series $F\left( x \right) = \sum\nolimits_{j \ge r} {a_j x^j / j!}$ gives $F\left( x \right) = e^{x}-x-1$. And therefore the generated potential polynomials are
\[
\left( {\frac{{x^2 / 2!}}{{e^x  - x - 1}}} \right)^z  = \left( {\frac{1}{2}\frac{{x^2 }}{{e^x  - x - 1}}} \right)^z  = \sum\limits_{j \ge 0} {G_j^{\left( z \right)} \frac{{x^j }}{{j!}}} .
\]
According to Howard's theorem we find
\begin{equation}
G_n^{\left( z \right)}  = \sum\limits_{i = 0}^n {\left( { - 1} \right)^i \binom{z + i - 1}{i}\binom{z + n}{n - i}2^{i}\frac{{n!i!}}{{\left( {n + 2i} \right)!}}B_{n + 2i,i} \left( {0, 1 ,1, \ldots } \right)} .
\end{equation}
Now we derive an expression for the exponential Bell polynomials $B_{n,i}\left(0,1,1,\ldots\right)$ in terms of the Stirling numbers of the second kind:
\begin{align*}
i!\sum\limits_{n \ge 0} {B_{n,i} \left( {0, 1, 1 , \ldots } \right)\frac{{x^n }}{{n!}}} & = \left( {F\left( x \right)} \right)^i  = \left( {e^x  - x - 1} \right)^i \\
& = \left( { - x + \sum\limits_{l \ge 1} {\frac{{x^l }}{{l!}}} } \right)^i  = \sum\limits_{j = 0}^i {\binom{i}{j}\left( { - 1} \right)^{i - j} x^{i - j} \left( {\sum\limits_{l \ge 1} {\frac{{x^l }}{{l!}}} } \right)^j } \\
& = \sum\limits_{j = 0}^i {\binom{i}{j}\left( { - 1} \right)^{i - j} x^{i - j} j!\sum\limits_{n \ge 0} {S\left( {n,j} \right)\frac{{x^n }}{{n!}}} } \\
& = \sum\limits_{n \ge 0} {\sum\limits_{j = 0}^i {\binom{i}{j}\left( { - 1} \right)^{i - j} j!S\left( {n,j} \right)\frac{{x^{n + i - j} }}{{n!}}} }\\
& = i!\sum\limits_{n \ge 0} {\left\{ {\frac{{n!}}{{i!}}\sum\limits_{j = 0}^i {\binom{i}{j}\left( { - 1} \right)^{i - j} j!} \frac{{S\left( {n - i + j,j} \right)}}{{\left( {n - i + j} \right)!}}} \right\}\frac{{x^n }}{{n!}}} .
\end{align*}
Hence
\begin{equation}
B_{n,i} \left( {0,1,1 , \ldots } \right) = \frac{{n!}}{{i!}}\sum\limits_{j = 0}^i {\binom{i}{j}\left( { - 1} \right)^{i - j} j!} \frac{{S\left( {n - i + j,j} \right)}}{{\left( {n - i + j} \right)!}}.
\end{equation}
Thus we obtain
\begin{equation}
G_n^{\left( z \right)}  = \sum\limits_{i = 0}^n {\binom{z + i - 1}{i}\binom{z + n}{n - i}2^{i}n!\sum\limits_{j = 0}^i {\binom{i}{j}\left( { - 1} \right)^j j!} \frac{{S\left( {n + i + j,j} \right)}}{{\left( {n + i + j} \right)!}}} .
\end{equation}
Substituting $z=k+1/2$ and $n=2k$ into this expression yields
\begin{equation}
G_{2k}^{\left( k+\frac{1}{2} \right)}  = \sum\limits_{i = 0}^{2k} {\binom{k + i - 1/2}{i}\binom{3k + 1/2}{2k - i}2^{i}\left(2k\right)!\sum\limits_{j = 0}^i {\binom{i}{j}\left( { - 1} \right)^j j!} \frac{{S\left( {2k + i + j,j} \right)}}{{\left( {2k + i + j} \right)!}}},
\end{equation}
hence by (\ref{pf4}) we finally have
\begin{equation}
a_k  = \frac{\left(2k\right)!}{2^k k!}\sum\limits_{i = 0}^{2k} {\binom{k + i - 1/2}{i}\binom{3k + 1/2}{2k - i}2^{i}\sum\limits_{j = 0}^i {\binom{i}{j}\left( { - 1} \right)^j j!} \frac{{S\left( {2k + i + j,j} \right)}}{{\left( {2k + i + j} \right)!}}}.
\end{equation}
This completes the proof of the theorem.
\end{proof}

\section{Acknowledgments}
I am grateful to Lajos L\'{a}szl\'{o}, who drew my attention to the paper of Brassesco and M\'{e}ndez. I also would like to thank the referees for their valuable remarks that help to improve the initial version of this paper.


\begin{thebibliography}{1}
\setlength{\itemsep}{2pt}

\bibitem{ref1}
C.~M.~Bender and S.~A.~Orszag,
\newblock {\em Advanced Mathematical Methods for Scientists and Engineers},
\newblock McGraw-Hill Book Company, 1978, p.\ 218.

\bibitem{ref2}
F.~T.~Howard, A theorem relating potential and Bell polynomials,
{\em Discrete Math.\/}~{\bf 39} (1982), 129--143.

\bibitem{ref3}
G.~Marsaglia and J.~C.~Marsaglia, A new derivation of Stirling's approximation to $n!$,
{\em Amer. Math. Monthly\/}~{\bf 97} (1990), 826--829.

\bibitem{ref4}
J.~M.~Borwein and R.~M.~Corless, Emerging tools for experimental mathematics,
{\em Amer. Math. Monthly\/}~{\bf 106} (1999), 899--909.

\bibitem{ref5}
L.~Comtet,
\newblock {\em Advanced Combinatorics: The Art of Finite and Infinite Expansions},
\newblock D. Reidel Publishing Company, 1974, p.\ 267.

\bibitem{ref6}
R.~B.~Paris and D.~Kaminski,
\newblock {\em Asymptotics and Mellin--Barnes Integrals},
\newblock Cambridge University Press, 2001, p.\ 32.

\bibitem{ref7}
S.~Brassesco and M.~A.~M\'{e}ndez, The asymptotic expansion for $n!$ and Lagrange inversion formula,
\newblock \href{http://arxiv.org/abs/1002.3894}{\tt http://arxiv.org/abs/1002.3894}.

\end{thebibliography}

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\noindent 2010 {\it Mathematics Subject Classification}:
Primary 11B65; Secondary 11B73, 41A60.

\noindent \emph{Keywords: } 
asymptotic expansions, factorial, Stirling coefficients, Stirling's
formula, Stirling numbers.


\bigskip
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\noindent (Concerned with sequences
\seqnum{A001163},
\seqnum{A001164},
\seqnum{A008277},
\seqnum{A050211}, and
\seqnum{A059022}.)

\bigskip
\hrule
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\vspace*{+.1in}
\noindent
Received March 15 2010;
revised version received  June 17 2010.
Published in {\it Journal of Integer Sequences}, June 21 2010.

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\noindent
Return to
\htmladdnormallink{Journal of Integer Sequences home page}{http://www.cs.uwaterloo.ca/journals/JIS/}.
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