*Alternative Representation for Binomials and Multinomies and Coefficient Calculation DOI: http://dx.doi.org/10.5772/intechopen.91422*

**Table 1.**

 *Layout pattern of the representation for the first nine binomial*

 *coefficients.*

X 1 X 0 1

*i*¼1 *ii* " #

X *δ* ∗ 0 1

8 >< >:

X *δ* ∗ 1 1

*δδ* <sup>∗</sup> <sup>1</sup> <sup>¼</sup><sup>1</sup>

0 B@

8 >< >:

⋯ • *δ* X ∗ ð Þ� *k*þ1 2

X 1

*j*¼1

ð Þ� *k*þ Y 1 1

*f*¼1

þ1k*k*

X *δ* ∗ *f*�1

<sup>¼</sup> ð Þ *<sup>x</sup>*<sup>1</sup> <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> *xk* <sup>þ</sup> *xk*þ<sup>1</sup> *<sup>n</sup>*

and *n*. This table is shown below.

*δ* ∗ *<sup>f</sup>* ¼0

**Appendix B: Table**

2 4

X 1

*j*¼1

ð Þ� *k*þ Y 1 1

*f*¼1

*δ* ∗ ð Þ� *<sup>k</sup>*þ<sup>1</sup> <sup>2</sup>�*<sup>δ</sup>* <sup>∗</sup>

¼

*δ* ∗ *<sup>f</sup>*�1�*<sup>δ</sup>* <sup>∗</sup> *f* � �

¼

**112**

*δ* ∗ ð Þ� *<sup>k</sup>*þ<sup>1</sup> <sup>1</sup>¼<sup>0</sup>

ð Þ� *k*þ1 1 � �

0 B@ *δδ* <sup>∗</sup> <sup>0</sup> <sup>¼</sup><sup>1</sup> *j*,*k*,*::*, *δδ* <sup>∗</sup> *k* , … , *δδ* <sup>∗</sup> *k*�1

> *δ* ∗ 1

> > <sup>2</sup> <sup>þ</sup><sup>1</sup>

*δδ* <sup>∗</sup> <sup>1</sup> <sup>¼</sup><sup>1</sup>

> *δ* ∗ 2

*δδ* <sup>∗</sup> <sup>2</sup> <sup>¼</sup><sup>1</sup>

X *δ* ∗ ð Þ� *k*þ1 2 1

8 >>><

0

BBB@

þ1k*k*

X *δ* ∗ *f*�1

> X 0 1

*i*¼1 *ii* " #

> *δ* ∗ *f*�1 *δ* ∗ *f*

! !

*δ* ∗ *<sup>f</sup>* ¼0

>>>:

*δδ* <sup>∗</sup> ð Þ� *k*þ1 2 ¼1

X 0 1

*i*¼1 *ii* " #

> 8 >>><

0

BBB@

>>>:

X *δ* ∗ *f*�1

*δδ* <sup>∗</sup> *f*�1 ¼1

1

*j*,*k*,*::*, *δδ* <sup>∗</sup> *f* , … , *δδ* <sup>∗</sup> *f*�1

> *x δ* ∗ *<sup>f</sup>*�1�*<sup>δ</sup>* <sup>∗</sup> *f*

*f*

This result confirms the hypothesis and the validity for (2).

<sup>1</sup> <sup>þ</sup><sup>1</sup>

⋯ X

*δ* ∗ <sup>0</sup> �*<sup>δ</sup>* <sup>∗</sup> ð Þ <sup>1</sup> <sup>þ</sup>1k*δδ* <sup>∗</sup>

⋯ X

*δ* ∗ <sup>1</sup> �*<sup>δ</sup>* <sup>∗</sup> ð Þ <sup>2</sup> <sup>þ</sup>1k*δδ* <sup>∗</sup> 9 >=

*Mathematical Theorems - Boundary Value Problems and Approximations*

*δ* ∗ *k*�1 *k* � � � � ⋯ � � ⋯ �

*δ* ∗ <sup>0</sup> �*<sup>δ</sup>* <sup>∗</sup> ð Þ <sup>1</sup> <sup>þ</sup>1k*<sup>k</sup>*

X 1

*j*¼1

þ1k*δδ* <sup>∗</sup> ð Þ� *k*þ1 1 þ1

> 9 >=

>; *γ δ* ∗ ð Þ� *<sup>k</sup>*þ<sup>1</sup> <sup>2</sup>�*<sup>δ</sup>* <sup>∗</sup>

> 3 7 <sup>5</sup>• *<sup>x</sup> δ* ∗ ð Þ� *k*þ1 1 *k*þ1 � �

⋯ X

*δ* ∗ *<sup>f</sup>*�1�*<sup>δ</sup>* <sup>∗</sup> *f* � �

*δ* ∗ <sup>1</sup> �*<sup>δ</sup>* <sup>∗</sup> ð Þ <sup>2</sup> <sup>þ</sup>1k*<sup>k</sup>*

⋯ ••• ⋯••• ⋯ ••• ⋯••• ⋯

*δ* ∗ ð Þ� *k*þ1 1

*δδ* <sup>∗</sup> ð Þ� *k*þ1 1 ¼1

, … , *δδ* <sup>∗</sup> ð Þ� *k*þ1 2

> *δ* ∗ *f*

*δδ* <sup>∗</sup> *f* ¼1

9 >=

1

CA *x δ* ∗ *<sup>f</sup>*�1�*<sup>δ</sup>* <sup>∗</sup> *f*

*f*

(where it was substituted with ð Þ *<sup>x</sup>*<sup>1</sup> <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> <sup>⋯</sup> <sup>þ</sup> *xk <sup>n</sup>* according to the hypothesis).

Next on **Table 1**, the symbolic representations for the binomial coefficients are presented. The value actually comes from the output of the program *coef.mc* provided in Section 3.2, after being evaluated with the corresponding values of *k*

>;

3 5• *x δ* ∗ ð Þ� *k*þ1 1 *k*þ1 � �

þ1k*δδ* <sup>∗</sup> *<sup>f</sup>* <sup>þ</sup><sup>1</sup> X 1 X 0 1

*i*¼1 *ii* " #

X 0 1

*i*¼1 *ii* " # *j*,*k*,*::*, *δδ* <sup>∗</sup>

*j*,*k*,*::*, *δδ* <sup>∗</sup>

⋯ X

*δ* ∗ ð Þ� *<sup>k</sup>*þ<sup>1</sup> <sup>2</sup>�*<sup>δ</sup>* <sup>∗</sup> ð Þ� *k*þ1 1 � �

ð Þ� *k*þ1 1 ð Þ� *<sup>k</sup>*þ<sup>1</sup> <sup>2</sup> *<sup>φ</sup>*

2

*k*¼1

þ1k*l*1

⤸

*δ* ∗ ð Þ� *k*þ1 2 ð Þ� *k*þ1 1 � � �

<sup>1</sup> , … , *δδ* <sup>∗</sup> 0

> 9 >= >; *γ δ* ∗ <sup>1</sup> �*<sup>δ</sup>* <sup>∗</sup> 2 <sup>2</sup> •⋯

<sup>2</sup> , … , *δδ* <sup>∗</sup> 1

þ1k*l*

1

1

CCA � <sup>⋯</sup>

(41)

1

CCA

CCA⋯

*<sup>k</sup>*¼<sup>1</sup> <sup>⤸</sup>

2

9 >= >; *γ δ* ∗ <sup>0</sup> �*<sup>δ</sup>* <sup>∗</sup> 1 <sup>1</sup> •

*j*¼1

>; *γ δ* ∗ *<sup>k</sup>*�1�*<sup>δ</sup>* <sup>∗</sup> *k <sup>k</sup>*�<sup>1</sup> *<sup>φ</sup>*

⋯ X 2

⋯ X 2

*k*¼1

⋯ X

ð Þ� *k*þ1 1

⋯ X

*δ* ∗ *<sup>f</sup>*�1�*<sup>δ</sup>* <sup>∗</sup> *f* � �

*δ* ∗ ð Þ� *<sup>k</sup>*þ<sup>1</sup> <sup>2</sup>�*<sup>δ</sup>* <sup>∗</sup> ð Þ� *k*þ1 1 � �

*j*, *k*,*::*, *δδ* <sup>∗</sup>

*δ* ∗ <sup>1</sup> �*<sup>δ</sup>* <sup>∗</sup> ð Þ <sup>2</sup> <sup>þ</sup>1k*<sup>l</sup>*

*k*¼1

*δ* ∗ <sup>0</sup> �*<sup>δ</sup>* <sup>∗</sup> ð Þ <sup>1</sup> <sup>þ</sup>1k*<sup>l</sup>*

*j*¼1

<sup>¼</sup> <sup>X</sup>*<sup>δ</sup>* <sup>∗</sup> 0 *δ* ∗ <sup>1</sup> ¼0

X*δ* ∗ 1 *δ* ∗ <sup>2</sup> ¼0

*δ* ∗ *<sup>k</sup>*�1�*<sup>δ</sup>* <sup>∗</sup> ð Þ *<sup>k</sup>* <sup>þ</sup>1k*<sup>k</sup>*

)

*Mathematical Theorems - Boundary Value Problems and Approximations*

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*DOI: http://dx.doi.org/10.5772/intechopen.91422*

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*Alternative Representation for Binomials and Multinomies and Coefficient Calculation*

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