Coefficient of $1$ in the expansion of $\left(1+x+\frac{1}{x}\right)^n$

asymptoticsbinomial-coefficientsmultinomial-theorem

What is the coefficient of $1$ in the expansion of $(1+x+\frac{1}{x})^n$? In other words, what is the sum of the coefficients of $x^ky^k$ in the expansion of $(1+x+y)^n$? Here, $n$ is a positive integer. To be specific, the coefficient is
\begin{equation*}
c_n=\sum_{k=0}^{\left[\frac{n}{2}\right]}\binom{n}{2k}\binom{2k}{k}.
\end{equation*}
If there is no explicit formula, an asymptotic formula with some analysis is also very fine and appreciated. Thanks.

Best Answer

A probabilistic approach: consider the random variable $X$ taking values on $\{-1,0,1\}$ with equal probability. Then its GF is $G_X(x)=\frac13 (1 + x + x^{-1})$.

Also, $E[X]=0$ and $\sigma^2_X=E[X^2]=\frac23$

Now, let $Y= \sum_{i=1}^N X_i$ where $X_i$ are iid with same distribution as above. Then $E[Y]=0$ and $\sigma^2_Y=N \frac{2}{3}$

Also $P(Y=0)=3^{-N} g_0$ where $g_0$ is the independent term in $(1 + x + x^{-1})^N$

But for large $N$, using CLT:

$$ P(Y=0) \approx \frac{1}{\sqrt{2 \pi \sigma^2_Y}}=\sqrt{\frac{3}{4 \pi N}}$$

Hence $$g_0 \approx 3^N \sqrt{\frac{3}{4 \pi N}}=3^{N+\frac12} \frac{1}{2 \sqrt{\pi N}}$$

This first order asymptotic approximation could be refined (eg).

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