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 A114604 #14 Jul 15 2024 10:17:58
%S A114604 1,5,43,709,23003,1481957,190305691,48796386661,25003673060507,
%T A114604 25613941912987493,52467767892904362139,214929296497738201165669,
%U A114604 1760788099067877263041671323,28849467307107603960961499533157
%N A114604 Numerator of partial sums of A005329/A006125.
%C A114604 To win a game, you must flip n+1 heads in a row, where n is the total number of tails flipped so far. The probability of having won before n+1 tails (that is, winning by flipping n+1 or fewer heads in a row) is a(n)/A006125(n). The probability of winning for the first time after n tails (that is, by flipping n+1 heads in a row) is A005329(n)/A006125(n).
%H A114604 Michael De Vlieger, <a href="/A114604/b114604.txt">Table of n, a(n) for n = 0..80</a>
%H A114604 Johann Cigler, <a href="https://arxiv.org/abs/2407.05768">Hankel determinants of backward shifts of powers of q</a>, arXiv:2407.05768 [math.CO], 2024. See p. 4.
%F A114604 a(n) = numerator(Sum_{k=0..n} A005329(k)/A006125(k)).
%F A114604 a(n) = a(n-1) * 2^(n+1) + A005329(n).
%e A114604 a(3) = 43 because 1/2 + 1/8 + 3/64 = 43/64, or because a(2) * 2^(2+1) + A005329(2) = 5 * 8 + 3 = 43.
%t A114604 Nest[Append[#1, #1[[-1]]*2^(#2 + 1) + Product[2^i - 1, {i, #2}]] & @@ {#, Length[#]} &, {1}, 13] (* _Michael De Vlieger_, Jul 15 2024 *)
%Y A114604 Cf. A005329, A006125.
%K A114604 easy,frac,nonn
%O A114604 0,2
%A A114604 _Joshua Zucker_, Dec 14 2005