<s>[INST] coefficient $x ^ n$ in development The advisor asks to verify that the coefficient of
$$x^n$$
in the development of:
$$(1+x)^{2n}+x(1+x)^{2n−1}+x2(1+x)^{2n−2}+......+x^n(1+x)^n$$
is equal to
$$\binom{2n+1}{n}$$
I tried for summations but not.
I did with the denominator change too, but I can not even,how can i match the 2 questions
 [/INST] You can explicitly evaluate your sum and extract the coefficient afterward.  Recall that  $$(y-x) (y^n + xy^{n-1} + \cdots + x^{n-1}y + x^n) = y^{n+1} - x^{n+1}$$  Letting $y = 1+x$ (so $y-x = 1$) gives
$$(1+x)^n + x(1+x)^{n-1} + \cdots + x^{n-1}(1+x) + x^n = (1+x)^{n+1} - x^{n+1}.$$
Then multiplying by $(1+x)^n$ gives your expression
$$(1+x)^{2n} + x(1+x)^{2n-1} + \cdots + x^n (1+x)^n = (1+x)^{2n+1} - (1+x)^n x^{n+1}.$$
The coefficient of $x^n$ of $(1+x)^nx^{n+1}$ is $0$, while the coefficient of $x^n$ in $(1+x)^{2n+1}$ is $${ 2n+1 \choose n}$$ by the binomial theorem.
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