For the past 25 years, I have wondered if this formula and algorithm are something interesting, or if they are already well know, but I have never been able to find any reference to them. I am hoping that someone on this forum might be able to help me answer this question, or might be able to refer me to someone who can.

The formula that I devised for generating the nth member of the sequence (starting counting at 0) for power p is:

p(n^(p-1))+((p(p-1))/2!)(n^(p-2))+((p(p-1)(p-2))/3!)(n^(p-3))+...+((p(p-1)(p-2)....(p-(p-1)))/p!)(n^(p-p))

Using this formula:

For p = 1, the sequence is 1,1,1,1,....
For p = 2, the sequence is 1,3,5,7,....
For p = 3, the sequence is 1,7,19,37,....
For p = 4, the sequence is 1,15,65,175,....

Any help with determining whether the use of this formula is already well known for generate the sequence of numbers for any power p, would be greatly appreciated.

f(n, p) = (n + 1)^p - n^p,

so it telescopes into the p^th powers of successive integers:

f(0, p) + f(1, p) + ... + f(n - 1, p) = n^p

because

(1)^p - (0)^p +
(2)^p - (1)^p +
(3)^p - (2)^p +
...
(n)^p - (n-1)^p =
(n)^p