– Chandrahas M. Halai
This is in continuation of my earlier articles
https://chandrahasblogs.wordpress.com/2021/07/30/how-many-laddoos/
https://chandrahasblogs.wordpress.com/2021/08/02/aryabhatts-sum-of-sums/
In the previous article I had discussed about the sum of sums. Let me start this article with the sum of sums of sums. That is, we want to find:
We get:
These are called the Pentatope numbers. They are the sum of Tetrahedral numbers. The fifth pentatope number is the sum of first five tetrahedral numbers. That is:
The above formula (i) and a general formula for sums of higher orders were derived by Naryan Pandit in his work Ganitakaumudi (1356 CE).
How about adding pentatope numbers? Sums of pentatope numbers are called 5-simplex numbers. Sums of sequences of natural numbers are in general called the Simplex numbers. Are we going to find formula for 5-simplex numbers and higher order sums recursively, like the way we have found the formulas for tetrahedral and pentatope numbers? Well, that will be a very cumbersome way to do it. Let us have a look at an alternative way of doing it.
Let me begin by making a table as shown table 1. First let us fill up column 0 with 1s. Now, put in cell of row1 and column 1, the sum of the cells above and above-left of it. That is the sum of columns 1 and 0 of row 0. Now, in the cell of row 2 and column 1, put the sum of the cells above and above-left of it. That is the sum of columns 1 and 0 of row 1 which 1 + 1 = 2. Now, in the cell of row 3 and column 1, put the sum of the cells above and above-left of it. That is the sum of columns 1 and 0 of row 2 which 1 + 2 = 3. In the same way, we fill up all the cells of column 1. And we see that column 1 contains the sequence of natural numbers. The nth number of column 1 is the sum of n 1s of column 0.
|
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
The number in cell of row 5 and column 3, is the sum of numbers in columns 3 and 2 of row 4, which is 4 + 6 = 10. You can fill the rest of the cells of the remaining columns in the same way. That is each cell is filled with the sum of numbers in the cells above and above-left.
Table 1 is actually left justified Pingala’s Meru Prastaar
(मेरु प्रस्तार). Refer my earlier article:
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. Refer table 2.
Table 2
The number in the cell of row n and column r is given by
Hence, we have
This generic formula for sums of higher orders was derived by Naryan Pandit in his work Ganitakaumudi (1356 CE).
Thus the fifth pentatope number is given by:
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