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

1, 1, 4, 22, 140, 969, 7084, 53820, 420732, 3782992, 32389076, 275617830, 2350749914, 20140518790, 173429992350, 1500850805160, 14550277251918, 133009333771170, 1198324107797254

Offset

0,3

Comments

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.

See the Munafo web page for complete description.

a(n) diverges from A002293 because Pi is not exactly 3.

Links

Table of n, a(n) for n=0..18.

Robert Munafo, Related to a Game of Chance

"My Math Forum" discussion thread, I give, duz... what is it?

"duz" blog entry, Random Walking (broken link)

"duz" blog entry, Random Walking, Sep 23 2008 (archived by R. Munafo on Dec 21 2010)

emath.ac.cn ("Mathematics Research and Development Network"), "Probability issues in random walks" (in Chinese)

Example

Numerators of series sum 1/2 + 1/32 + 4/512 + 22/8192 + 140/131072 + ...

Crossrefs

Sequence in context: A240586 A369156 A002293 * A369126 A003287 A077056

Adjacent sequences: A181781 A181782 A181783 * A181785 A181786 A181787

Keyword

nonn,frac

Author

Robert Munafo, Dec 21 2010

Extensions

a(18) from Robert Munafo, Dec 22 2010

Corrected and added links by Robert Munafo, Jan 01 2024

Status

approved