Coefficient of xk- 0- k- n in expansion of P-1-1-x-1-x2-1-xn

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Binomial Coefficients

Coefficient o...

Question

Coefficient of $x^{k}, (0 \leq k \leq n)$ in expansion of $P = 1 + (1 + x) + (1 + x)^{2} . . . . . . + (1 + x)^{n}$

^{n} C_{k}

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^{n + 1} C_{n - k - 1}

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^{n} C_{n - k}

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^{n + 1} C_{k + 1}

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x^{k} (0 \leq k \leq n)

1 + (1 + x) + (1 + x)^{2} + \dots + (1 + x)^{n}

The coefficient of $x^{m}$ in $(1 + x)^{k} + (1 + x)^{k + 1} + . . . . . . . . . . . . . . + (1 + x)^{n},$ $(k \leq m \leq n)$ , is given by:

1 + (1 + x) + (1 + x)^{2} + . . . . + (1 + x)^{n}

x^{k} (0 \leq k \leq n)

^{(k - 1)} C_{(k - 1)} +^{k} C_{(k - 1)} +^{(k + 1)} C_{(k - 1)} + . . . . . . . +^{(n + k - 2)} C_{(k - 1)}

= ?

n > 1

1 + x + x^{2} + . . . . + x^{n - 1} = 0

n \sum k = 0 (x^{k} + \frac{1}{x^{k}})

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