This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A370399 #21 Mar 21 2024 20:50:00
%S A370399 1,1,1,2,1,1,3,3,1,1,3,18,6,1,1,15,60,15,10,1,1,15,2,100,25,20,1,1,15,
%T A370399 210,75,1050,105,35,1,1,15,5880,5880,73500,29400,588,70,1,1,135,30240,
%U A370399 35280,529200,1852200,15435,588,126,1,1,135,340200,453600,7938000,466754400,3111696,1481760,19845,252,1
%N A370399 Triangle read by rows: T(n, k) is the denominator of the probability of winning a certain game while playing optimally.
%C A370399 T(n, k) is the numerator of the probability of winning a game M(n, k) while playing optimally. The game is played by a single player who begins with n tokens of which k are blue and the rest are red; 0 <= k <= n > 0. Each move consists of randomizing the order of the remaining tokens, observing their resulting order, choosing a number m of tokens to remove, and removing the m tokens that are ordered last; m must be at least 1 but no more than half of the remaining tokens. Play continues until only one token remains; the game is won if that token is blue, otherwise the game is lost.
%C A370399 Let Pr(n,k) be the probability of winning a game M(n,k). Then Pr(n+1,1) = (n/(n-1))*Pr(n,1) if n is a power of 2, Pr(n,1) otherwise. So lim_{n->oo} Pr(n,1) = (1/2)*(2/3)*(4/5)*(8/9)*(16/17)*... = A083864.
%e A370399 The values of Pr(n,k) begin as follows:
%e A370399 .
%e A370399 n\k| 0 1 2 3 4 5 6 7
%e A370399 ---+---------------------------------------------------------
%e A370399 1 | 0/1 1/1
%e A370399 2 | 0/1 1/2 1/1
%e A370399 3 | 0/1 1/3 2/3 1/1
%e A370399 4 | 0/1 1/3 11/18 5/6 1/1
%e A370399 5 | 0/1 4/15 31/60 11/15 9/10 1/1
%e A370399 6 | 0/1 4/15 1/2 69/100 21/25 19/20 1/1
%e A370399 7 | 0/1 4/15 101/210 49/75 829/1050 94/105 34/35 1/1
%e A370399 ...
%e A370399 We can calculate Pr(4,2) using the table below, given the values of Pr(n,k) for n=3 and for n=2. The leftmost column lists each of the six possible results of randomizing the n=4 tokens during the first move; in each randomized sequence, the red and blue tokens are represented by "r" and "b", respectively.
%e A370399 .
%e A370399 randomized probability result if result if
%e A370399 sequence of last 1 token last 2 tokens
%e A370399 of tokens occurrence is removed are removed
%e A370399 ========== =========== ============== =============
%e A370399 rrbb 1/6 Pr(3,1) = 1/3 Pr(2,0) = 0/1
%e A370399 rbrb 1/6 Pr(3,1) = 1/3 Pr(2,1) = 1/2
%e A370399 brrb 1/6 Pr(3,1) = 1/3 Pr(2,1) = 1/2
%e A370399 rbbr 1/6 Pr(3,2) = 2/3 Pr(2,1) = 1/2
%e A370399 brbr 1/6 Pr(3,2) = 2/3 Pr(2,1) = 1/2
%e A370399 bbrr 1/6 Pr(3,2) = 2/3 Pr(2,2) = 1/1
%e A370399 .
%e A370399 For example, when we get rbrb it's better to remove the last two tokens (one r and one b) instead of removing only the last token (b). So the probability of winning M(4,2) is
%e A370399 Pr(4,2) = (1/6)(1/3) + (1/6)(1/2) + (1/6)(1/2) + (1/6)(2/3) + (1/6)(2/3) + (1/6)(1/1) = 11/18.
%e A370399 Of course Pr(n,k) >= k/n, because k/n could be achieved by removing 1 token on each move.
%Y A370399 Numerators are in A370398.
%K A370399 nonn,frac,tabl
%O A370399 1,4
%A A370399 _Julian Zbigniew Kuryllowicz-Kazmierczak_, Feb 17 2024
%E A370399 More terms from _Jon E. Schoenfield_, Feb 24 2024