Hypercube elements

J, 13 bytes

[:p.2&^;$&_.5

Inspired by @alephalpha's PARI/GP answer. Try it online with J.js.

Background

By the binomial theorem,

Thus, the output for input n consist precisely of the coefficients of the above polynomial.

Code

[:p.2&^;$&_.5  Monadic verb. Argument: n

        $&_.5  Yield an array of n instances of -0.5.
    2&^        Compute 2^n.
       ;       Link the results to the left and right.
               This specifies a polynomial of n roots (all -0.5)
               with leading term 2^n.  
[:p.           Convert from roots to coefficients.

Samau, 8 5 bytes

Saved 3 bytes thanks to Dennis.

▌2\$ⁿ

Hex dump (Samau uses CP737 encoding):

dd 32 2f 24 fc

Explanation:

▌        read a number
 2\      push the array [1 2]
   $     swap
    ⁿ    take the convolution power

Convolving two vectors is equivalent to multiplying two polynomials. Similarly, taking the n-th convolution power is equivalent to taking the n-th power of a polynomial.

MATL, 8 bytes

1i:"2:X+

Inspired by @alephalpha's PARI/GP answer.

Try it online! (uses Y+ for modern day MATL)

How it works

1        % Push 1.
 i:      % Push [1 ... input].
   "     % Begin for-each loop:
    2:   %   Push [1 2].
      X+ %   Take the convolution product of the bottom-most stack item and [1 2].