✞ Might as well ask me a question if you're here stalking my page
512
Ting · 24d
Hihi Julia, what's your favourite book? ^^
Simp #7263 · 25d
What languages do you speak
Simp #7263 · 1mo
Why is there a gun on your profile (I'm NOT your simp)
Yami · 2mo
Why are you on Retrospring if you should be studying?
My Intuition told me that you would be here.
Julia · 2mo
i can ask myself a question? wtf
Yami · 2mo
What are your thoughts on Super Mario being an Italian with Japanese parents?
Are you jealous?
Yami · 2mo
Solve this:
Let p(n) denote the number of distinct partitions of a positive integer n.
For example:
p(4) equals 5 because the partitions of 4 are: 4, 3+1, 2+2, 2+1+1, and 1+1+1+1.
Now consider the following sum:
S(N) = sum from k=1 to N of [(-1)^(k+1) * k * p(N-k)]
Find the closed-form expression for S(N) in terms of N.
Prove that S(N) is always divisible by N for all positive integers N greater than or equal to 1.
Generalize the problem for any positive integer M, and determine if the sum S_M(N) = sum from k=1 to N of [(-1)^(k+1) * k^M * p(N-k)] is divisible by a function of N and M. Prove your result.