# Can this sum be simplified?

Can this expression be further simplified : $\sum_{k=0}^{n-1} { n -1 \choose k } (-2)^{k} (2n - k)!$? This is the coefficient of $x^{2n}$ in the formal power series expansion of $(1-2x)^{n-1} \times \sum_{ k \geq 0} k! x^k$.

Motivation: I came across this when trying to solve a problem using inclusion-exclusion principle, I am not mentioning the original problem because I am interested in this sum as an independent problem.

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Have you looked at the OEIS? – Robin Chapman Oct 6 '10 at 19:03
@robin: what is OEIS – anonymous Oct 6 '10 at 19:06
@robin: yes. No luck : research.att.com/~njas/sequences/index.html?q=2,12,336,17760 – ACARCHAU Oct 6 '10 at 19:09
Did you try WolframAlpha? It gives a solution in terms of the "Kummer confluent hypergeometric function", which may or may not be any better. – Rahul Narain Oct 6 '10 at 19:13
@Rahul: Thanks for suggesting wolfram alpha, I will take a look. – ACARCHAU Oct 6 '10 at 19:15

It's a confluent hypergeometric function $\rm\ (2n)! \phantom{f}_1 F_1(1-n;\: -2\:n;\: -2)\$, which can be expressed in closed form in terms of modified Bessel functions of the first kind, namely

$$\begin{split}(-1)^n &2^{\frac12-n}e^{-1-i\pi n}(n+1)\Gamma\left(\frac12-n\right)\Gamma(2n)I_{\frac12(-2n-1)}(1)-\\&(-1)^n 2^{\frac12-n}e^{-1-i\pi n}\Gamma\left(\frac12-n\right)\Gamma(2n)I_{\frac12(1-2n)}(1)\end{split}$$

It may well have a simpler closed form, since the various hypergeometric simplifiers in CAS are not always optimal (above is via Mathematica). It is not known by the OEIS Superseeker, so you may wish to submit it there.

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As a way of restating Bill's answer, what you have can in fact be expressed in terms of the so-called (generalized) Bessel polynomial:

$$y_n(x;a)=(n+a-1)_n \left(\frac{x}{2}\right)^n {}_1 F_1 \left(-n;-2n-a+2;\frac{2}{x}\right)$$

(the Kummer hypergeometric series degenerates to a polynomial here because both the numerator and denominator parameters are negative integers)

Your original expression, then, in terms of the (generalized) Bessel polynomial, is

$$(-2)^{n-1}(n+1)!y_{n-1}(-1;4)$$

The references in the DLMF can point you to papers where the (generalized) Bessel polynomials have been studied; for an elementary treatment, see Chihara's An Introduction to Orthogonal Polynomials.

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