# [Python 2], 140 bytes

<!-- language-all: lang-python -->

 n=input()
 x=[[0,]*n+[1]+n*[0,]]
 z=n%2
 exec'x+=[[(i%2^z)*sum(x[-1][i-1:i+2])for i in range(2*n+1)]];z^=1;'*(n-1)
 print map(sum,zip(*x))[1:-1]

[Try it online!] or [Try all test cases](https://tio.run/nexus/python2#PcxBCoMwEEDRvaeYjTgTE@ikdBPxJCEFaWOZhVOxCsHLW7vp8sPnHc88wohKoYLSx3ixyWgbObVqfpEq2HutfQW55EdT2vNBqf19J/PZJizRcYriOEjrE43vBQREYRn0ldGfFlNK3X7vuWsMqmOqYF5EV1DbgIPGTsOMJ2V3mdEUosjhNI8/JTpvK1IYUeiIbL292ptlTl8 "Python 2 – TIO Nexus")

[Python 2]: https://docs.python.org/2/
[Try it online!]: https://tio.run/nexus/python2#FcxBCoMwEEDRfU6RjTiTKHSyVHKSMIIUW2bhNFiFkMvbdPnh826Novk6AU2JKT0GduoTsVf3DzY1ahfMVrZnX3w7QLqwVHTfa4eSRuIkI03iA@Prc1ixovZY9b1BaBIh81yXSHPvQEdCkw/R0@5rhiYMVTK4gphoatR9E/0A "Python 2 – TIO Nexus"

<sup>For `n=3`</sup> 
Starts with a list with `n` zeros surround an one - `[[0, 0, 0, 1, 0, 0, 0]]` 
Generate the full pyramid

 [[0, 0, 0, 1, 0, 0, 0],
 [0, 0, 1, 0, 1, 0, 0],
 [0, 1, 0, 2, 0, 1, 0]]

Rotate 90º and sum each row, discarding the first and the last one (only zeros)

 [[0, 0, 0],
 [0, 0, 1],
 [0, 1, 0],
 [1, 0, 2],
 [0, 1, 0],
 [0, 0, 1],
 [0, 0, 0]]