I am interested in approximating the sum of the squares of the multinomial coefficients, i.e.


$a\_\ell^p := \sum\_{k\_0+\ldots+k\_p = \ell} (\frac{\ell!}{k\_0! \ldots k\_p!})^2$ 


or more general, 


$a\_\ell^{\alpha\_0,\ldots, \alpha\_p} := \sum\_{k\_0+\ldots+k\_p = \ell} (\prod\_{i=0}^p \alpha\_i^{k\_i})^2(\frac{\ell!}{k\_0! \ldots k\_p!})^2$ 


Here $p$ is prime, and $\ell$ is an integer smaller then $p$, $k\_i$ are non-negative integers. I would like to obtain some simple expression in terms of $\ell$ and $p$, a good approximation for large $p$, an upper bound will be good. 


I saw some results on recursive formula for such expressions, but not estimates. Should I just go with a Stirling formula in all the terms or is there something better done in this direction? 


Does someone knows what would be the Mathematica/Maple code for calculating such sums as functions from $p$ and $l$?


Thanks!



Estimate on sum of squares of multinomial coefficients
Douglas already commented that the asymptotics for fixed $p$ and $l\to \infty$ shoudl follow from standard methods. One gets
$$a\_{\ell}^p\approx (p+1)^{2\ell+\frac{p+1}{2}}(4\pi \ell)^{-\frac{p}{2}}.$$
See theorem 4 in ["Counting Abelian squares"](http://arxiv.org/abs/0807.5028), by Richmond and Shallit. Notice that these numbers appear also in combinatorics when considering abelian squares, or more generally abelian powers, on a fixed alphabet.


For the asymptotics that you're interested in, at least in the unweighted case, one can say
$$a \_{\ell} ^p=\sum \_{j=0} ^{\ell} \binom{p}{j}\sum \_{a \_1+ \cdots +a \_j = \ell \atop a \_i \geq 1} \binom{\ell}{a \_1,a \_2,\dots,a \_j}^2$$
which makes it clear that $a \_{\ell}^p$ is a polynomial in $p$ of fixed degree $\ell$. The coefficient of $\binom{p}{\ell}$ is $(\ell!)^2$, and the coefficient of $\binom{p}{\ell -1}$ is $\frac{\ell-1}{4}(\ell!)^2$, so you have
$$a \_{\ell}^p =\ell!p^{\ell}-\ell!\frac{\ell(\ell-1)}{4}p^{\ell-1}+O(p^{\ell-2}).$$


