Pascal's Column Sums
MATL, 16 bytes
tZv=Gq:&quot;t5BZ+]vs

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Explanation
This repeatedly applies convolution to generate the rows. For example, for input n=5 we start with the first row
0 0 0 0 1 0 0 0 0

Convolving with [1 0 1] gives
0 0 0 1 0 1 0 0 0

Repeating the operation gives
0 0 1 0 2 0 1 0 0

then
0 1 0 3 0 3 0 1 0

etc. Concatenating these arrays vertically and computing the sum of each column gives the result.
t       % Input n implictly. Duplicate
Zv      % Symmetric range. Gives [1 2 3 4 5 4 3 2 1] for input 5
=       % Equal to (element-wise). Gives [0 0 0 0 1 0 0 0 0]. This is the first row
Gq:     % Push [1 2 ... n-1]
&quot;       % For each. This executes the following code n-1 times
  t     %   Duplicate
  5B    %   Push 5 in binary, that is, [1 0 1]
  Z+    %   Convolution keeping size
]       % End
v       % Concatenate all results vertically 
s       % Sum. Display implicitly.