When you want to get the square of a binomial of two variables x and y, you will have:

$(x+y{)}^2=x^2+2xy+y^2(x+y)^2\; =\; x^2\; +\; 2xy\; +\; y\; ^2$

And the cube:

$(x+y{)}^3=x^3+3x^{2}y+3x{y}^2+y^3(x+y)^3\; =\; x^3\; +\; 3x^2y\; +\; 3xy^2\; +y^3$

It is known from many centuries ago that for an exponent n, the result of a binomial x + y raised to the n-th power is:

Or using the sumation notation:

Each coefficient of a term has the following value:

Each coefficient value coincides with the amount of combinations without replacements of a set of n elements using only k different ones of that set.

Let's see the total sum of the coefficients of the different powers for the binomial:

$(x+y{)}^0(1)(x+y)^0(1)$

$(x+y{)}^1=x+y(2)(x+y)^1\; =\; x+y(2)$

$(x+y{)}^2=x^2+2xy+y^2(4)(x+y)^2\; =\; x^2\; +\; 2xy\; +\; y\; ^2(4)$

$(x+y{)}^3=x^3+3x^{2}y+3x{y}^2+y^3(8)(x+y)^3\; =\; x^3\; +\; 3x^2y\; +\; 3xy^2\; +y^3(8)$

Task

Create a function that returns (an array) of the coefficients sums from 0 to n (inclusive), where the last element is the sum of all previous elements.

We add some examples below:

for n = 0, return 1, 1
for n = 1, return 1, 2, 3
for n = 2, return 1, 2, 4, 7
for n = 3, return 1, 2, 4, 8, 15

Features of the test

Low Performance Tests
Number of tests = 50
9 < n < 101

High Performance Tests
Number of Tests = 50
99 < n < 5001

N.B. In C, input is limited to 0 <= n <= 63