# Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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by C. C. Chen and D. E. Daykin
Proc. Amer. Math. Soc. 59 (1976), 394-398 Request permission

## Abstract:

A cascade $C$ is defined as a sum of binomial coefficients $C = \left ( {\begin {array}{*{20}{c}} {{a_h}} \\ h \\ \end {array} } \right ) + \left ( {\begin {array}{*{20}{c}} {{a_{h - 1}}} \\ {h - 1} \\ \end {array} } \right ) + \cdots + \left ( {\begin {array}{*{20}{c}} {{a_t}} \\ t \\ \end {array} } \right )$ where ${a_h} > {a_{h - 1}} > \cdots > {a_t}$. In this expression, we assume that $(_h^a) = 0$ whenever $a < h$. Given a cascade $C$ and a sequence $\varepsilon = \langle {\varepsilon _h},{\varepsilon _{h - 1}}, \ldots ,{\varepsilon _t}\rangle$ of signs (i.e. ${\varepsilon _i} = + 1\;{\text {or}}\; - 1$ for each $i$), we define $\varepsilon C = {\varepsilon _h}\left ( {\begin {array}{*{20}{c}} {{a_h}} \\ h \\ \end {array} } \right ) + \cdots + {\varepsilon _t}\left ( {\begin {array}{*{20}{c}} {{a_t}} \\ t \\ \end {array} } \right ).$ Also, we put $\alpha C = \left ( {\begin {array}{*{20}{c}} {{a_h}} \\ {h + 1} \\ \end {array} } \right ) + \left ( {\begin {array}{*{20}{c}} {{a_{h - 1}}} \\ h \\ \end {array} } \right ) + \cdots + \left ( {\begin {array}{*{20}{c}} {{a_t}} \\ {t + 1} \\ \end {array} } \right ).$ In the paper, we prove that for any sequence $\langle {n_0}, {n_1}, \ldots ,{n_s}\rangle$ of integers, there exist a cascade $C$ and a corresponding sequence $\varepsilon$ of signs such that ${n_i} = \varepsilon {\alpha ^i}C$ for $i = 0,\;1, \ldots ,s$ where ${\alpha ^0}C = C,\;{\alpha ^1}C = \alpha C,\;{\alpha ^2}C = \alpha ({\alpha ^1}C)$, and recursively, ${\alpha ^n}C = \alpha ({\alpha ^{n - 1}}C)$.
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