Same Number Pattern

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 $\frac{e^{x}}{1+x}$ also appears in the following formulae:


MATH

MATH

MATH

MATH

MATH


MATH

MATH

MATH

MATH

MATH

  1. Suppose that MATH Then
    MATH

    MATH

    MATH

    MATH

    MATH

Exercise

Let MATH

  1. Find the general form of the n-th derivative $f^{(n)}(x).$

  2. Find the recursive formula associated with $f^{(n)}(x).$

  3. How is the above recursive formula related to the matrix?

  4. Generate the coefficients occurred in $f^{(n)}(x)$ for $n=1,2,\cdots ,50$ with the spreadsheet.