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

Author

Robert Munafo, Dec 21 2010

Keywords

• nonn
• frac

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.

Examples

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

Links

• 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)

Extensions

a(18) from Robert Munafo, Dec 22 2010

Corrected and added links by Robert Munafo, Jan 01 2024