[seqfan] Re: a new sequence idea

Max Alekseyev maxale at gmail.com
Sun Feb 12 20:35:16 CET 2023

Numerators and denominators for the diagonal T(n,n) are given by
https://oeis.org/A322755 and https://oeis.org/A322756 respectively.


On Thu, Feb 9, 2023 at 10:04 PM Yifan Xie <xieyifan4013 at 163.com> wrote:

> Here is Problem 6 (generalized version) in 2023 AIME I:
> There is a random sequence of binary digits containing n 0's and m 1's.
> Then a person uses the best strategy to guess the digits one-by-one and
> every digit will be revealed after the person guesses it. Find the
> mathematical expectation of the number of digits the person guesses
> correctly.
> Suppose that at a certain time the remaining sequence unrevealed consists
> of j 0's and k 1's, than the person will guess 0 if j>=k, or 1 if j<=k so
> the possibility that the person guesses correctly is max{j, k}/j+k. Next we
> should calculate the possibility that the remainder contains such digits,
> which can be described as C(n-j, n+m-j-k)*C(j, j+k)/C(n, n+m).
> However things are actually very complicated and I just gave up trying to
> develop a simple formula (it is possible for me to write a c++ program).
> Another problem is that although the sequence terms are all fractions,
> T(n, m)'s denominator is proved to be a divisor of C(n, n+m). Would it be
> better to represent the numerator of the sequence (when the denominator is
> C(n, n+m)), or to represent the ordinary numerator and denominator in two
> separate sequences?
> Please help.
> Best regards,
> Yifan Xie (xieyifan4013 at 163.com)
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