# 1. Use procedure prodmake in C25.txt to conjecture a product formula for the exponential function. Use human ingenuity to prove it. U := 40: for u from 1 to U do: prodmake(add(q^i/i!,i=0..u),q,u,list): od: # Sample Output: [-1, 1/2, 1/3, 0, 1/5, -1/6, 1/7, 0, 0, -1/10, 1/11, 0, 1/13, -1/14, -1/15, 0, 1/17, 0, 1/19, 0, -1/21, -1/22, 1/23, 0, 0, -1/26]. A # few patterns emerge: # - Any n with repeated prime factors has exponent 0 # - The sign of each exponent is determined by the parity (odd -> +1, even -> -1) of the number of prime factors n has. # This sounds like the (negative of the) Moebius function to me. The factors in the product expansion would then be, I guess, (1 - q^n)^(-mu(n)/n) # Proof: Take a log of both sides. We then must show that q = sum((-mu(n)/n)*log(1-q^n),n=1..infinity). We may expand the logarithms log(1-x), # and group together by denominator/power of q sums of mobius functions. Using the identity that sum over the divisors of n of the mobius is 1 or # 0 (when n is 1 or > 1, respectively), the claim follows. # 2. Procedure vecptns(n) gives the set of triples ("vector partitions") [Q,P1,P2] where Q is a partition into distinct parts, and P1, P2 are # partitions such that all the parts of Q,P1,P2, add up to n. Is the sequence "number of vector partitions on n" in the OEIS? Is it for that reason? # If it is for another reason, prove that they are actually the same for all n, not just the first 30 terms. # First ten terms: 3,8,19,41,83,161,299,538,942,1610,… # …which is related to the sequence sigma(2n)/n (sigma = sigma_1 is the sum of divisors function). As for how to prove the relation, I will have to # study vector partitions more.