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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Representation of numbers by cascades


Authors: C. C. Chen and D. E. Daykin
Journal: Proc. Amer. Math. Soc. 59 (1976), 394-398
MSC: Primary 05A17
MathSciNet review: 0414385
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Abstract: A cascade $ C$ is defined as a sum of binomial coefficients

$\displaystyle C = \left( {\begin{array}{*{20}{c}} {{a_h}} \\ h \\ \end{array} }... ... \cdots + \left( {\begin{array}{*{20}{c}} {{a_t}} \\ t \\ \end{array} } \right)$

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

$\displaystyle \varepsilon C = {\varepsilon _h}\left( {\begin{array}{*{20}{c}} {... ...silon _t}\left( {\begin{array}{*{20}{c}} {{a_t}} \\ t \\ \end{array} } \right).$

Also, we put

$\displaystyle \alpha C = \left( {\begin{array}{*{20}{c}} {{a_h}} \\ {h + 1} \\ ... ... + \left( {\begin{array}{*{20}{c}} {{a_t}} \\ {t + 1} \\ \end{array} } \right).$

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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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1976-0414385-X
PII: S 0002-9939(1976)0414385-X
Article copyright: © Copyright 1976 American Mathematical Society