Here's an interesting polynomial pattern I picked up.

Sum[n, k=1](1) = n

Sum[n, k=1](2k-1) = n^2

Sum[n, k=1](3k^2-3k+1) = n^3

Sum[n, k=1](4k^3-6k^2+4k-1) = n^4

Sum[n, k=1](5k^4-10k^3+10k^2-5k+1) = n^5

Sum[n, k=1](6k^5-15k^4+20k^3-15k^2+6k-1) = n^6

Can you see the pattern?

.....

Well, although I'm having a big fat load of trouble nesting this polynomial pattern, it can be understood this way: n^x = Sum[n, k=1](Natural(n) - Triangle(n) + Tetrahedral(n) -... +/- 1), where Natural(n) is the nth natural number, Triangle(n) is the nth triangle number, and so on. (I'll explain what triangle numbers and tetrahedral numbers are in a second. {Not to offend those who already know what they are.})

The "..." in this formula represents an extension of the natural, triangle, tetrahedral... sequence. Natural numbers can be written in this way:

The nth natural number = n

Triangle numbers can be written in this way:

The nth triangle number = (n(n+1))/2

Tetrahedral numbers can be written in this way:

The nth tetrahedral number = (n(n+1)(n+2))/6

The relationship between these sequences is such- the nth number in the kth kind of number sequence (where natural numbers correspond to k=0, triangles correspond to k=1, and so on) is equal to Product[k, i=0](n+i)/[(k+1)!]

I'm sure I can use combinations in the nesting process instead of this series, (since combinations are closely related to Pascal's Triangle, where these numbers are directly taken from), but I haven't yet figured out how.

Any form of assistance is appreciated.

Logs of Negative Numbers

Happy New Year!

Well, after taking a short break for the holidays I'm back on track with a new strange discovery. I noticed that the infinite sum Sum(infinity, n=0) {1/(e^n)} = (1+rt(5))/2, or the golden ratio. (While trying to evaluate the sum with raw computing power on my calculator I noticed that the terms completely stopped growing at around n=80 or so.) I can't seem to come up with any reason that e and the golden ratio would be related, and the constant I reached could be close to the value of the golden ratio simply by coincidence. (This actually seems pretty likely because the when the terms stopped growing the constant was slightly lower than the golden ratio- 1.581976707. {The golden ratio is equal to 1.618033989.})

Any ideas?