2023.10.30 闲话

Another Side .

I got no time - The Living Tombstone

I got no time

I got no time to live

I got no time to live

And I can't say goodbye

And I'm regretting having memories

Of my friends who they used to be

Beside me before they left me to die

And I know this is

I know this is the truth

'Cause I've been staring at my death so many times

These scary monsters roaming in the halls

I wish I could just block the doors

And stay in bed until the clock will chime

So my flashlight's on, and stay up 'til dawn

I got this headache and my life's on the line

I felt like I won, but I wasn't done

The nightmare repeats itself every time

Got to keep my calm, and carry on

Stay awake until the sun will shine

But I'm not so strong, and they're not gone

They're still out there to take what's left of mine

I have this urge

I have this urge to kill

I have this urge to kill and show that I'm alive

I'm getting sick from these apologies

From people with priorities

That their life matters so much more than mine

But I'm stuttering

I'm stuttering again

No one will listen and no one will understand

Because I'm crying as much as I speak

'Cause no one likes me when I shriek

Want to go back to when it all began

So my flashlight's on, and stay up 'til dawn

I got this headache and my life's on the line

I felt like I won, but I wasn't done

The nightmare repeats itself every time

Got to keep my calm, and carry on

Stay awake until the sun will shine

But I'm not so strong, and they're not gone

They're still out there to take what's left of mine

从 NaCly_Fish 那看到个好的，能直接刻画 OGF 的二项前缀和（？这啥名）. 原文链接 .

断言：

\[[z^n]\dfrac1{1-z}F\left(\dfrac z{1-z}\right)=\sum_{k=0}^n\dbinom nkf_k \]

证明只需要直接代入：

\[\begin{aligned}\,[z^n]\dfrac1{1-z}F\left(\dfrac z{1-z}\right)&=[z^n]\sum_{k\ge 0}f_k\cdot\dfrac{z^k}{(1-z)^{k+1}}\\&=\sum_{k=0}^nf_k\cdot[z^{n-k}]\dfrac1{(1-z)^{k+1}}\\&=\sum_{k=0}^n\dbinom nkf_k\end{aligned} \]

例：求

\[\sum_{i=1}^n\dbinom ni^2 \]

组合数的 OGF \(F(z)=(1+z)^n\) . 那么就有：

\[\mathrm{ans}=[z^n]\dfrac1{1-z}\left(1+\dfrac z{1-z}\right)^n=[z^n]\dfrac1{(1-z)^{n+1}}=\dbinom{2n}n \]

小练习：求

\[\sum_{k=0}^n\dbinom nif_i \]

其中 \(f\) 是 Fibonacci 数列（\(f_0=1\)）.

推歌：Catch My Breath - Kelly Clarkson .