By Chandrahas M. Halai
This is in continuation of my earlier articles
https://chandrahasblogs.wordpress.com/2022/05/24/choice-yes-or-no-halayudhas-combinatorial-relation/
and
https://chandrahasblogs.wordpress.com/2021/08/05/naryan-pandits-sum-of-sums-of-sums-of-sums/.
|
Col 0 |
Col 1 |
Col 2 |
Col 3 |
Col 4 |
Col 5 |
Col 6 |
Col 7 |
Col 8 |
Col 9 |
|
| Row 0 |
1 |
|||||||||
| Row 1 |
1 |
1 |
||||||||
| Row 2 |
1 |
2 |
1 |
|||||||
| Row 3 |
1 |
3 |
3 |
1 |
||||||
| Row 4 |
1 |
4 |
6 |
4 |
1 |
|||||
| Row 5 |
1 |
5 |
10 |
10 |
5 |
1 |
||||
| Row 6 |
1 |
6 |
15 |
20 |
15 |
6 |
1 |
|||
| Row 7 |
1 |
7 |
21 |
35 |
35 |
21 |
7 |
1 |
||
| Row 8 |
1 |
8 |
28 |
56 |
70 |
56 |
28 |
8 |
1 |
|
| Row 9 |
1 |
9 |
36 |
84 |
126 |
126 |
84 |
36 |
9 |
1 |
Table 1
Let us consider the left justified Meru Prastaar as shown in Table 1.
The column 2 contains the sequence of triangular numbers. The nth number of column 2 is the sum of first n numbers of column 1, that is, the sum of first n natural numbers. The 5th number of column 2 is the sum of first 5 numbers (first 5 natural numbers) of column 1, which is 1 + 2 + 3 + 4 + 5 = 15.
The column 3 contains the sequence of tetrahedral numbers. The nth number of column 3 is the sum of first n numbers of column 2, that is, the sum of first n triangular numbers. The 4th number of column 3 is the sum of first 4 numbers (first 4 triangular numbers) of column 2, which is 1 + 3 + 6 + 10 = 20.
The column 4 contains the sequence of pentatope numbers. The nth number of column 4 is the sum of first n numbers of column 3, that is, the sum of first n tetrahedral numbers. The 5th number of column 4 is the sum of first 5 numbers (first 5 tetrahedral numbers) of column 3, which is 1 + 4 + 10 + 20 +35 = 70.
All these sums of sequence of natural numbers are in general called simplex numbers. The numbers in column 1, that is, the sequence of natural numbers are called 1-simplex numbers. The numbers in column 2, that is, the sequence of triangular numbers are called 2-simplex numbers. The numbers in column 3, that is, the sequence of tetrahedral numbers are called 3-simplex numbers. The numbers in column 4, that is, the sequence of pentatope numbers are called 4-simplex numbers. And so on. The numbers in column r are the r-simplex numbers.
The r-simplex number is denoted by Sr. The nth r-simplex number is denoted by Sr(n).
The fifth pentatope number is S4(5) = 70.
Sr(n) is the number in the cell of rth column and (r + n – 1)th row.
The numbers in the cells of Meru Prastaar are binomial coefficients.
Hence, we have
But how do we prove that the nth number of column r is the sum of first n numbers of column (r – 1)?
We have Halayudha’s combinatorial relation:
We have
Also, we have
Substitute this in (1), we have
In the (r – 1)th column of Meru Prastaar the numbers start from (r – 1)th row. Hence, the first n numbers in the (r – 1)th column are from row numbers (r – 1) to (r + n – 2).
We have
Hence, we have
Which is to be proved.
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