I'm looking for a closed form expression for the sum
$P_n(x) =\sum_{0\leq k\leq n/2}\binom{n}{k}x^k$,
where $n$ is a given positive integer and $k$ runs over nonnegative integers between $0$ and $n/2$. The first such polynomials are
$1,1+2x,1+3x,1+4x+6x^2,1+5x+10x^2,…$
Alternatively one might want to subtract the half of the last term $\binom{n}{n/2}x^{n/2}$ when $n$ is even to obtain the polynomials
$Q_1(x) = 1, Q_2(x) = 1+x, Q_3(x) = 1+3x, Q_4(x) = 1 + 4x + 3x^2,\ldots$
These have the nice property that $Q_n(x) + x^n Q_n(1/x) = (1+x)^n$, but I am unable to obtain a closed form for either sequence of functions. One can obtain a recursive formula and also a closed form expression for the generating function
$G(x,y) = \sum_{n=0}^{\infty} Q_n(x)y^n$,
but this form involves a term of the type
$\frac{\sqrt{1- 4y}}{1 – ay},$
which I am unable to expand as a power series in $y$. If you find an expression for the sequences $P_n$ or $Q_n$ or are able to express the above function as a power series in $y$, I would be very grateful. Thanks!
Best Answer
$\sum_{0\leq k\leq n/2}\binom{n}{k}x^k = (x+1)^n - \binom{n}{\left\lfloor\frac{n}{2}\right\rfloor + 1} x^{\left\lfloor\frac{n}{2}\right\rfloor + 1} {_2F_1}\left(1,\left\lfloor\frac{n}{2}\right\rfloor - n + 1;\left\lfloor \frac{n}{2}\right\rfloor + 2;-x\right)$,
where $_2F_1$ is the hypergeometric function.