Question

Find the original generating function in closed form for (1,2,3,4,5,...).

Answer 1

since 1,1,1,1,1... has its function

∞
∑x^n = 1/(1-x)
n=0

take the derivative

∑nx^(n-1) = 1/(1-x)^2

now replace n-1 with n and you have

∑(n+1)x^n = 1/(1-x)^2
generates
1,2,3,4,5,...