Comparision of power of sequence of all natural number and the factorial of

Ideas
1

Some time ago I have found correlation between power of natural numbers and factorial.
I do not know if this is a known fact or not, but I would like to know why is this so.
For example if you have a sequence of all natural numbers and you calculate the second, third or any other (n) power of this number, and you afterwards calculate the difference between results and recalculate differences between this results the same time of the given power you get factorial of the n.
Maybe is better if you look at the photos of calculation because English is not my mother language.

power-2.png
power-2.png989×481 3.18 KB

power-3.png
power-3.png936×481 3.91 KB

power-4.png
power-4.png936×481 4.86 KB

power-5.png
power-5.png936×481 5.44 KB

etc. it goes on and on

https://www.facebook.com/valter.zikovic/media_set?set=a.10202811116581713.1073741832.1314896407&type=3

2

power-6.png
power-6.png936×481 6.28 KB

power-7.png
power-7.png940×481 6.94 KB

power-8.png
power-8.png940×481 7.88 KB

power-9.png
power-9.png940×481 8.22 KB

3

The basic reason is that the n-th derivative of x^n is constant and equal to the factorial of n.

In[2]:= D[x^9, {x, 9}]

Out[2]= 362880

In[3]:= 9!

Out[3]= 362880

Then the observed result is a special case of the mean value theorem for divided differences (Wikipedia) for a polynomial function and equidistant interpolation nodes, after taking into account the relationship between finite differences and divided differences. The proof follows from the Newton-form divided difference representation of the Lagrange interpolating polynomial.

2 Likes
4

Same phenomenon occur due to special equation from binomial transform (Ballu Equation).