cp's OEIS Frontend

A181784 Numerators of a series sum related to a game of chance.

%I A181784 #29 Jan 02 2024 04:52:31
%S A181784 1,1,4,22,140,969,7084,53820,420732,3782992,32389076,275617830,
%T A181784 2350749914,20140518790,173429992350,1500850805160,14550277251918,
%U A181784 133009333771170,1198324107797254
%N A181784 Numerators of a series sum related to a game of chance.
%C A181784 Consider a 1-dimensional random walk from 0 with equal-probability steps of Pi and -1.  One way to compute the probability of eventually walking below 0 is as the sum over n of the probabilities of becoming negative after a walk with exactly n steps of Pi (n >= 0) and max(ceiling(n*Pi),1) steps of -1.  The total number of walks of such length for a given n is 2^(n+max(ceiling(n*Pi),1)), or 2^(n+A004084(n)) (n >= 1), forming a sequence of denominators, and this sequence gives the numerators, the number of possible sequences of length (n+max(ceiling(n*Pi),1)) drawn from {Pi, -1} such that no partial sum except the total sum is < 0.
%C A181784 See the Munafo web page for complete description.
%C A181784 a(n) diverges from A002293 because Pi is not exactly 3.
%H A181784 Robert Munafo, <a href="http://mrob.com/pub/math/seq-a181784.html">Related to a Game of Chance</a>
%H A181784 "My Math Forum" discussion thread, <a href="https://mathforums.com/t/i-give-duz-what-is-it.4121/">I give, duz... what is it?</a>
%H A181784 "duz" blog entry, <a href="http://zdu.spaces.live.com/blog/cns!C95152CB25EF2037!127.entry"> Random Walking</a> (broken link)
%H A181784 "duz" blog entry, <a href="http://mrob.com/pub/seq/duz-20080923.pdf">Random Walking</a>, Sep 23 2008 (archived by R. Munafo on Dec 21 2010)
%H A181784 emath.ac.cn ("Mathematics Research and Development Network"), <a href="https://bbs.emath.ac.cn/thread-331-1-1.html">"Probability issues in random walks"</a> (in Chinese)
%e A181784 Numerators of series sum 1/2 + 1/32 + 4/512 + 22/8192 + 140/131072 + ...
%K A181784 nonn,frac
%O A181784 0,3
%A A181784 _Robert Munafo_, Dec 21 2010
%E A181784 a(18) from _Robert Munafo_, Dec 22 2010
%E A181784 Corrected and added links by _Robert Munafo_, Jan 01 2024