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%I #29 Jan 02 2024 04:52:31
%S 1,1,4,22,140,969,7084,53820,420732,3782992,32389076,275617830,
%T 2350749914,20140518790,173429992350,1500850805160,14550277251918,
%U 133009333771170,1198324107797254
%N Numerators of a series sum related to a game of chance.
%C 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 See the Munafo web page for complete description.
%C a(n) diverges from A002293 because Pi is not exactly 3.
%H Robert Munafo, <a href="http://mrob.com/pub/math/seq-a181784.html">Related to a Game of Chance</a>
%H "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 "duz" blog entry, <a href="http://zdu.spaces.live.com/blog/cns!C95152CB25EF2037!127.entry"> Random Walking</a> (broken link)
%H "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 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 Numerators of series sum 1/2 + 1/32 + 4/512 + 22/8192 + 140/131072 + ...
%K nonn,frac
%O 0,3
%A _Robert Munafo_, Dec 21 2010
%E a(18) from _Robert Munafo_, Dec 22 2010
%E Corrected and added links by _Robert Munafo_, Jan 01 2024
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