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probname); ?>
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Problem

In the following exercise we study the $\textit{binomial coefficients }$ $\left( \begin{array}{c} n \\ k \end{array}\right)$.

a) First assume that $n$ and $k$ are integers such that $0 \le k \le n$. Suppose you didn't know anything about $\left( \begin{array}{c} n \\ k \end{array}\right)$, except the relations $\left( \begin{array}{c} n \\ k \end{array}\right) = \left( \begin{array}{c} n-1 \\ k-1 \end{array}\right) + \left( \begin{array}{c} n-1 \\ k \end{array}\right) \qquad \text{ and } \qquad \left( \begin{array}{c} n \\ 0 \end{array}\right) = 1 \, .$ Compute the generating function $B_n(x) := \sum_{ k \ge 0 } \left( \begin{array}{c} n \\ k \end{array}\right) x^k$ and use it to find a formula for $\left( \begin{array}{c} n \\ k \end{array}\right)$.

b) Convince yourself that your computation is identical if we allow $n$ to be any complex number and $k$ to be any nonnegative integer.

c) Compute the generating function $B(x) := \sum_{ n \ge 0 } \sum_{ k \ge 0 } \left( \begin{array}{c} n \\ k \end{array}\right) x^k y^n$ and use it to find a formula for the generating function $\beta_k(y) := \displaystyle \sum_{ n \ge k } \left( \begin{array}{c} n \\ k \end{array}\right) y^n$.

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Authors
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References
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Problem Sets This Problem Belongs to:
parent_nid; ?> Set:
VARIABLES
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DEFINITIONS
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