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%I #15 Feb 08 2021 11:46:42
%S 0,0,1,11,39,473,19897,63683,5731597,22926439,280212089,20175270749,
%T 224810160067,6294684482461,1321883741325001,1208579420640469,
%U 68486167169628137,17258514126746312369,178860964586279976467,6053755724458706915971,3305350625554453976644453
%N Numerator of the expected fraction of guests without a napkin in Conway's napkin problem with n guests.
%D Peter Winkler, Mathematical Puzzles: A Connoisseur's Collection, A K Peters, 2004, p. 122.
%H Anders Claesson and T. Kyle Petersen, <a href="http://arxiv.org/abs/math/0505080">Conway’s napkin problem</a>, arXiv:math/0505080 [math.CO], 2005.
%H Anders Claesson and T. Kyle Petersen, <a href="https://doi.org/10.1080/00029890.2007.11920408">Conway's napkin problem</a>, Amer. Math. Monthly 114 (No. 3, 2007), 217-231.
%H Niklas Eriksen, <a href="https://doi.org/10.1080/00029890.2008.11920556">The freshman's approach to Conway's napkin problem</a>, Amer. Math. Monthly 115 (No. 6, 2008), 492-498.
%H Steven R. Finch, <a href="http://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, arXiv:2001.00578 [math.HO], 2020, p. 2.
%F a(n)/A341233(n) = Sum_{k=2..n} (1-2^(2-k))/k!.
%F Lim_{n->oo} a(n)/A341233(n) = (2-sqrt(e))^2 (A248788).
%e 0, 0, 1/12, 11/96, 39/320, 473/3840, 19897/161280, 63683/516096, 5731597/46448640
%o (Python)
%o from sympy import numer, S, factorial
%o def A341232(n):
%o return numer(sum((1-S(2)**(2-k))/factorial(k) for k in range(2,n+1)))
%o (Python)
%o from math import factorial
%o from fractions import Fraction
%o def a(n):
%o s = sum(Fraction(2**k-4, 2**k*factorial(k)) for k in range(2, n+1))
%o return s.numerator
%o print([a(n) for n in range(1, 22)]) # _Michael S. Branicky_, Feb 07 2021
%Y Cf. A248788, A341233 (denominators).
%K nonn,frac
%O 1,4
%A _Pontus von Brömssen_, Feb 07 2021
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