A366568 P(n,n+2k-2) are the numerators of the probability that a gambler with initial capital n goes bankrupt exactly at round n+2k-2, randomly losing or winning 1 unit of money in each round, written as lower triangle T(n,k), n>=1, 1<=k<=n.

1, 1, 1, 1, 1, 1, 1, 3, 5, 5, 1, 1, 9, 7, 7, 1, 5, 7, 7, 21, 21, 1, 3, 5, 3, 45, 33, 33, 1, 7, 27, 75, 165, 297, 429, 429, 1, 1, 35, 55, 275, 143, 1001, 715, 715, 1, 9, 11, 77, 429, 1001, 1001, 429, 2431, 2431

Offset: 1

Author

Hugo Pfoertner, Oct 24 2023

Keywords

• nonn
• frac
• tabl

Examples

			The triangle begins
  1;
  1, 1;
  1, 1, 1;
  1, 3, 5, 5;
  1, 1, 9, 7,  7;
  1, 5, 7, 7, 21, 21;
  1, 3, 5, 3, 45, 33, 33;
The numbering is as follows, with
 [n, rounds, P] = [initial capital, coin flips, probability of ruin]:
  [1, 1,  1/2]
  [2, 2,  1/4] [1, 3,   1/8]
  [3, 3,  1/8] [2, 4,   1/8] [1, 5,  1/16]
  [4, 4, 1/16] [3, 5,  3/32] [2, 6,  5/64] [1, 7, 5/128]
  [5, 5, 1/32] [4, 6,  1/16] [3, 7, 9/128] [2, 8, 7/128] [1,  9,  7/256]
  [6, 6, 1/64] [5, 7, 5/128] [4, 8, 7/128] [3, 9, 7/128] [2, 10, 21/512]
                                                           [1, 11, 21/1024]
With initial capital odd, ruin can only occur at odd numbered rounds, with even initial capital only at even numbered rounds.
		

Links

• Louis Bachelier, Calcul des probabilités. Tome I, Gauthier-Villars, Paris, 1912.
• Eric Weisstein's World of Mathematics, Random Walk--1-Dimensional.
• Wikipedia, One-dimensional random walk

Crossrefs

A366569 are the corresponding denominators.

Cf. A086116, A141387.

Formula

P(m,mue) = (mue/m) * mue! / (((mue - m)/2)! * ((mue + m)/2)!) * (1/2)^mue, with round count mue and initial capital m (Bachelier, 1912, page 103).

P(m,mue) = 0 for mue < m and for mue + m odd.