find $\sum_{k=0}^{t}(-1)^k\binom{t}{k}^2$ for odd t then for even t

Question

find $$\sum_{k=0}^{t}(-1)^k\binom{t}{k}^2$$ for $t=2n$ then for $t=2n+1$

I tried by expand $(1-x)^n(1-x)^n$, with no result.

Any Help ?

Answer 1

It is better to exploit symmetry. Since $\binom{t}{k}=\binom{t}{t-k}$, we have: $$ \sum_{k=0}^{t}\binom{t}{k}^2(-1)^k = [x^t]\left[\left(\sum_{k=0}^{t}\binom{t}{k}(-1)^k x^k\right)\cdot\left(\sum_{k=0}^{t}\binom{t}{t-k}x^{t-k}\right)\right]$$ hence: $$ \sum_{k=0}^{t}\binom{t}{k}^2(-1)^k = [x^t]\left[(1-x)^t(1+x)^t\right]=[x^t](1-x^2)^t$$ so: $$ \sum_{k=0}^{t}\binom{t}{k}^2(-1)^k=\left\{\begin{array}{rcl}0&\text{if}&t\equiv 1\pmod{2},\\ \binom{t}{t/2}(-1)^{t/2}&\text{if}&t\equiv 0\pmod{2}.\end{array}\right.$$

Answer 2

$\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle} \newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack} \newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,} \newcommand{\dd}{{\rm d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,{\rm e}^{#1}\,} \newcommand{\fermi}{\,{\rm f}} \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,} \newcommand{\half}{{1 \over 2}} \newcommand{\ic}{{\rm i}} \newcommand{\iff}{\Longleftrightarrow} \newcommand{\imp}{\Longrightarrow} \newcommand{\pars}[1]{\left(\, #1 \,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\pp}{{\cal P}} \newcommand{\root}[2][]{\,\sqrt[#1]{\vphantom{\large A}\,#2\,}\,} \newcommand{\sech}{\,{\rm sech}} \newcommand{\sgn}{\,{\rm sgn}} \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert}$ $\ds{\sum_{k = 0}^{t}\pars{-1}^{k}{t \choose k}^{2}:\ {\large ?}}$.

\begin{align}&\color{#66f}{\large\sum_{k = 0}^{t}\pars{-1}^{k}{t \choose k}^{2}} =\sum_{k = 0}^{t}\pars{-1}^{k}{t \choose k} \oint_{\verts{z}\ =\ 1}{\pars{1 + z}^{t} \over z^{k + 1}}\,{\dd z \over 2\pi\ic} \\[3mm]&=\oint_{\verts{z}\ =\ 1}{\pars{1 + z}^{t} \over z} \sum_{k = 0}^{t}{t \choose k}\pars{-\,{1 \over z}}^{k}\,{\dd z \over 2\pi\ic} =\oint_{\verts{z}\ =\ 1}{\pars{1 + z}^{t} \over z} \bracks{1 + \pars{-\,{1 \over z}}}^{t}\,{\dd z \over 2\pi\ic} \\[3mm]&=\pars{-1}^{t}\oint_{\verts{z}\ =\ 1}{\pars{1 - z^{2}}^{t} \over z^{t + 1}} \,{\dd z \over 2\pi\ic} =\pars{-1}^{t}\sum_{k=0}^{t}{t \choose k}\pars{-1}^{k} \oint_{\verts{z}\ =\ 1}{1 \over z^{t - 2k + 1}}\,{\dd z \over 2\pi\ic} \\[3mm]&=\pars{-1}^{t}\sum_{k=0}^{t}{t \choose k}\pars{-1}^{k}\delta_{k,t/2} \end{align}

$$ \color{#66f}{\large\sum_{k = 0}^{t}\pars{-1}^{k}{t \choose k}^{2}} =\color{#c00000}{\large% \left\{\begin{array}{lcl} \pars{-1}^{t/2}{t \choose t/2} & \mbox{if} & t\ \mbox{is even} \\[2mm] 0 & & \mbox{otherwise} \end{array}\right.} $$