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A322632 Decimal expansion of the real solution to 23*x^5 - 41*x^4 + 10*x^3 - 6*x^2 - x - 1 = 0. Constant occurring in the asymptotic behavior of the number of lattice paths of slope 2/5, observed by D. Knuth.

%I A322632 #14 May 27 2019 12:35:28
%S A322632 1,6,3,0,2,5,7,6,6,2,9,9,0,3,5,0,1,4,0,4,2,4,8,0,1,8,4,9,3,1,5,9,8,6,
%T A322632 3,0,0,5,1,4,5,8,4,4,2,6,6,9,0,1,4,9,4,0,5,8,4,9,8,5,0,2,6,5,9,5,2,5,
%U A322632 6,8,9,1,2,9,8,6,8,5,0,4,7,9,8,3,4,1,3,2,4,1
%N A322632 Decimal expansion of the real solution to 23*x^5 - 41*x^4 + 10*x^3 - 6*x^2 - x - 1 = 0. Constant occurring in the asymptotic behavior of the number of lattice paths of slope 2/5, observed by D. Knuth.
%C A322632 In his 2014 lecture in Paris "Problems That Philippe (Flajolet) Would Have Loved" D. Knuth discussed as Problem 4 "Lattice Paths of Slope 2/5" and reported as an empirical observation that A[5*t-1,2*t-1]/B[5*t-1,2*t-1] = a - b/t + O(t^-2), with constants a~=1.63026 and b~=0.159. For the meaning of A and B see A322631. The exact values of a (this sequence) and b (A322633) were found in 2016 by Banderier and Wallner.
%H A322632 Cyril Banderier, Michael Wallner, <a href="https://arxiv.org/abs/1605.02967">Lattice paths of slope 2/5</a>, arXiv:1605.02967 [cs.DM], 10 May 2016.
%H A322632 D. E. Knuth, <a href="http://www-cs-faculty.stanford.edu/~uno/flaj2014.pdf">Problems That Philippe Would Have Loved</a>, Paris 2014.
%e A322632 1.6302576629903501404248018493159863005145844266901494058498502659525689...
%p A322632 evalf[100](solve(23*x^5-41*x^4+10*x^3-6*x^2-x-1=0,x)[1]); # _Muniru A Asiru_, Dec 21 2018
%t A322632 RealDigits[Root[23#^5 - 41#^4 + 10#^3 - 6#^2 - # - 1&, 1], 10, 100][[1]] (* _Jean-François Alcover_, Dec 30 2018 *)
%o A322632 (PARI) solve(x=1,2,23*x^5-41*x^4+10*x^3-6*x^2-x-1)
%Y A322632 Cf. A322631, A322633.
%K A322632 nonn,cons
%O A322632 1,2
%A A322632 _Hugo Pfoertner_, Dec 21 2018