Product of Head Counts
Problem 602
Alice enlists the help of some friends to generate a random number, using a single unfair coin. She and her friends sit around a table and, starting with Alice, they take it in turns to toss the coin. Everyone keeps a count of how many heads they obtain individually. The process ends as soon as Alice obtains a Head. At this point, Alice multiplies all her friends' Head counts together to obtain her random number.
As an illustration, suppose Alice is assisted by Bob, Charlie, and Dawn, who are seated round the table in that order, and that they obtain the sequence of Head/Tail outcomes THHH—TTTT—THHT—H beginning and ending with Alice. Then Bob and Charlie each obtain 2 heads, and Dawn obtains 1 head. Alice's random number is therefore .
Define to be the expected value of Alice's random number, where is the number of friends helping (excluding Alice herself), and is the probability of the coin coming up Tails.
It turns out that, for any fixed , is always a polynomial in . For example, .
Define to be the coefficient of in the polynomial . So , , and .
You are given that .
Find .