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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.

%I A366568 #10 Oct 29 2023 14:10:18
%S A366568 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,
%T A366568 75,165,297,429,429,1,1,35,55,275,143,1001,715,715,1,9,11,77,429,1001,
%U A366568 1001,429,2431,2431
%N 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.
%H A366568 Louis Bachelier, <a href="https://gallica.bnf.fr/ark:/12148/bpt6k9804939z/f125.item">Calcul des probabilités. Tome I</a>, Gauthier-Villars, Paris, 1912.
%H A366568 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RandomWalk1-Dimensional.html">Random Walk--1-Dimensional</a>.
%H A366568 Wikipedia, <a href="https://en.wikipedia.org/wiki/Random_walk#One-dimensional_random_walk">One-dimensional random walk</a>
%F A366568 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).
%F A366568 P(m,mue) = 0 for mue < m and for mue + m odd.
%e A366568 The triangle begins
%e A366568   1;
%e A366568   1, 1;
%e A366568   1, 1, 1;
%e A366568   1, 3, 5, 5;
%e A366568   1, 1, 9, 7,  7;
%e A366568   1, 5, 7, 7, 21, 21;
%e A366568   1, 3, 5, 3, 45, 33, 33;
%e A366568 The numbering is as follows, with
%e A366568  [n, rounds, P] = [initial capital, coin flips, probability of ruin]:
%e A366568   [1, 1,  1/2]
%e A366568   [2, 2,  1/4] [1, 3,   1/8]
%e A366568   [3, 3,  1/8] [2, 4,   1/8] [1, 5,  1/16]
%e A366568   [4, 4, 1/16] [3, 5,  3/32] [2, 6,  5/64] [1, 7, 5/128]
%e A366568   [5, 5, 1/32] [4, 6,  1/16] [3, 7, 9/128] [2, 8, 7/128] [1,  9,  7/256]
%e A366568   [6, 6, 1/64] [5, 7, 5/128] [4, 8, 7/128] [3, 9, 7/128] [2, 10, 21/512]
%e A366568                                                            [1, 11, 21/1024]
%e A366568 With initial capital odd, ruin can only occur at odd numbered rounds, with even initial capital only at even numbered rounds.
%Y A366568 A366569 are the corresponding denominators.
%Y A366568 Cf. A086116, A141387.
%K A366568 nonn,frac,tabl
%O A366568 1,8
%A A366568 _Hugo Pfoertner_, Oct 24 2023