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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, 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
1,8
LINKS
Louis Bachelier, Calcul des probabilités. Tome I, Gauthier-Villars, Paris, 1912.
Eric Weisstein's World of Mathematics, Random Walk--1-Dimensional.
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.
EXAMPLE
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.
CROSSREFS
A366569 are the corresponding denominators.
Sequence in context: A200334 A138112 A106233 * A198492 A077860 A261340
KEYWORD
nonn,frac,tabl
AUTHOR
Hugo Pfoertner, Oct 24 2023
STATUS
approved