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A366995 a(n) is the numerator of the expected end time of a game with three gamblers, one of which starts with capital n, the others with capital 1 each, conditional on the event that one of the two poor players wins.

%I A366995 #14 Nov 01 2023 10:01:13
%S A366995 3,39,359,2477,119667,1522705,46419629,6143100517,5472109127035,
%T A366995 790136773603303,278129286200597661,16684426086791338103,
%U A366995 503067648850136040148699,2626565018569118643191009,10920130209346850287269887104735,236686188450953790757840351941895
%N A366995 a(n) is the numerator of the expected end time of a game with three gamblers, one of which starts with capital n, the others with capital 1 each, conditional on the event that one of the two poor players wins.
%C A366995 In each round of the game, 1 unit is transferred from one randomly chosen player to another. Players play until they are out of money, so when the first player is out the other two continue to play. The winner is the player who ends up with all n+2 units of money.
%H A366995 Pontus von Brömssen, <a href="/A366995/b366995.txt">Table of n, a(n) for n = 1..53</a>
%H A366995 Persi Diaconis and Stewart N. Ethier, <a href="https://doi.org/10.1214/21-STS826">Gambler’s ruin and the ICM</a>, Statist. Sci. 37 (3) 289-305, August 2022.
%H A366995 Persi Diaconis, <a href="https://sites.math.rutgers.edu/~zeilberg/expmath/diaconis23.pdf">Gambler's ruin with k gamblers</a>, slides from talk in the Rutgers Experimental Mathematics Seminar, <a href="https://sites.math.rutgers.edu/~zeilberg/expmath/archive23.html">Fall 2023 Semester</a>, Oct 12, 2023.
%H A366995 Experimental Mathematics, <a href="https://vimeo.com/876882233">Gambler’s ruin with k gamblers</a>, recording of talk, Vimeo video, Oct 22, 2023.
%o A366995 (Sage)
%o A366995 from itertools import permutations
%o A366995 def T(n):
%o A366995     nodes = [(i,j) for i in range(n+2) for j in range((n+2-i)//2+1)]
%o A366995     m = len(nodes)
%o A366995     Q0 = {x:{y:0 for y in nodes} for x in nodes}
%o A366995     for x in nodes:
%o A366995         c1 = x+(n+2-sum(x),)
%o A366995         for i,j in permutations(range(3),int(2)):
%o A366995             if c1[i] and c1[j]:
%o A366995                 c2 = list(c1)
%o A366995                 c2[i] -= 1
%o A366995                 c2[j] += 1
%o A366995                 y = (c2[0],min(c2[1:]))
%o A366995                 if c2[0] != n+2:
%o A366995                     Q0[x][y] += n+2-c2[0]
%o A366995     Q0 = matrix(QQ,[list(R.values()) for R in Q0.values()])
%o A366995     s = sum(Q0.columns())
%o A366995     Q = identity_matrix(QQ,m-1)
%o A366995     for i in range(1,m):
%o A366995         for j in range(1,m):
%o A366995             if s[i] != 0: Q[i-1,j-1] -= Q0[i,j]/s[i]
%o A366995     return (Q**(-1)*ones_matrix(QQ,m-1))[-2,0]
%o A366995 def A366995(n):
%o A366995     return T(n).numerator()
%o A366995 def A366996(n):
%o A366995     return T(n).denominator()
%Y A366995 Cf. A366566 (a(n)/A366996(n) rounded to nearest integer), A366996 (denominators).
%K A366995 nonn,frac
%O A366995 1,1
%A A366995 _Pontus von Brömssen_, Oct 31 2023