Representation of numbers by cascades
Authors: Chen C. C. , Daykin D. E. 
Published: June 2010

A cascade

C
C

is defined as a sum of binomial coefficients
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper C equals StartBinomialOrMatrix a Subscript h Baseline Choose h EndBinomialOrMatrix plus StartBinomialOrMatrix a Subscript h minus 1 Baseline Choose h minus 1 EndBinomialOrMatrix plus midline-horizontal-ellipsis plus StartBinomialOrMatrix a Subscript t Baseline Choose t EndBinomialOrMatrix">

C
=

(

a
h

h

)

+

(

a

h
−
1

h
−
1

)

+
⋯
+

(

a
t

t

)

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

>
⋯
>

a
t

{a_h} > {a_{h - 1}} > \cdots > {a_t}

. In this expression, we assume that

(
h
a

)
=
0

(_h^a) = 0

whenever

a
>
h

a > h

. Given a cascade

C
C

and a sequence

ε
=
⟨

ε
h

,

ε

h
−
1

,
…
,

ε
t

⟩

\varepsilon = \langle {\varepsilon _h},{\varepsilon _{h - 1}}, \ldots ,{\varepsilon _t}\rangle

of signs (i.e.

ε
i

=
+
1

or

−
1

{\varepsilon _i} = + 1\;{\text {or}}\; - 1

for each

i
i

), we define
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="epsilon upper C equals epsilon Subscript h Baseline StartBinomialOrMatrix a Subscript h Baseline Choose h EndBinomialOrMatrix plus midline-horizontal-ellipsis plus epsilon Subscript t Baseline StartBinomialOrMatrix a Subscript t Baseline Choose t EndBinomialOrMatrix period">

ε
C
=

ε
h

(

a
h

h

)

+
⋯
+

ε
t

(

a
t

t

)

.

\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
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha upper C equals StartBinomialOrMatrix a Subscript h Baseline Choose h plus 1 EndBinomialOrMatrix plus StartBinomialOrMatrix a Subscript h minus 1 Baseline Choose h EndBinomialOrMatrix plus midline-horizontal-ellipsis plus StartBinomialOrMatrix a Subscript t Baseline Choose t plus 1 EndBinomialOrMatrix period">

α
C
=

(

a
h

h
+
1

)

+

(

a

h
−
1

h

)

+
⋯
+

(

a
t

t
+
1

)

.

\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

⟨

n
0

,

n
1

,
…
,

n
s

⟩

\langle {n_0},\,{n_1}, \ldots ,{n_s}\rangle

of integers, there exist a cascade

C
C

and a corresponding sequence

ε
\varepsilon

of signs such that

n
i

=
ε

α
i

C

{n_i} = \varepsilon {\alpha ^i}C

for

i
=
0
,

1
,
…
,
s

i = 0,\;1, \ldots ,s

where

α
0

C
=
C
,

α
1

C
=
α
C
,

α
2

C
=
α
(

α
1

C
)

{\alpha ^0}C = C,\;{\alpha ^1}C = \alpha C,\;{\alpha ^2}C = \alpha ({\alpha ^1}C)

, and recursively,

α
n

C
=
α
(

α

n
−
1

C
)

{\alpha ^n}C = \alpha ({\alpha ^{n - 1}}C)

.