The number pattern
1 | |||||||
1 | -1 | 0 | 0 | 0 | 0 | 0 | 0 |
1 | -2 | 2 | 0 | 0 | 0 | 0 | 0 |
1 | -3 | 6 | -6 | 0 | 0 | 0 | 0 |
1 | -4 | 12 | -24 | 24 | 0 | 0 | 0 |
1 | -5 | 20 | -60 | 120 | -120 | 0 | 0 |
1 | -6 | 30 | -120 | 360 | -720 | 720 | 0 |
1 | -7 | 42 | -210 | 840 | -2520 | 5040 | -5040 |
that appeared in the successive derivatives of
also appears in the following formulae:
The successive derivatives of
The integral formulae:
Suppose that
Then we have
Suppose that
Then
The integral formulae
Suppose that
Then
Let
Find the general form of the n-th derivative
Find the recursive formula associated with
How is the above recursive formula related to the matrix?
Generate the coefficients occurred in for with the spreadsheet.