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Decimal expansion of the real solution to 11571875*x^5 - 5363750*x^4 + 628250*x^3 - 97580*x^2 + 5180*x - 142 = 0, multiplied by 3/7. Constant occurring in the asymptotic behavior of the number of lattice paths of slope 2/5, observed by D. Knuth.
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%I #17 May 27 2019 12:15:19

%S 1,5,8,6,6,8,2,2,6,9,7,2,0,2,2,7,7,5,5,1,4,7,8,0,4,1,9,8,1,6,3,9,2,7,

%T 3,9,2,0,6,3,7,3,7,5,7,5,5,3,0,6,6,9,7,7,9,6,3,6,6,5,2,4,0,9,5,4,4,0,

%U 7,6,7,0,2,1,0,9,5,3,1,2,5,7,9,6,1,9,1,7,1,9,2

%N Decimal expansion of the real solution to 11571875*x^5 - 5363750*x^4 + 628250*x^3 - 97580*x^2 + 5180*x - 142 = 0, multiplied by 3/7. Constant occurring in the asymptotic behavior of the number of lattice paths of slope 2/5, observed by D. Knuth.

%C 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 (A322632) and b (this sequence) were found in 2016 by Banderier and Wallner.

%H 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 Don E. Knuth, <a href="http://www-cs-faculty.stanford.edu/~uno/flaj2014.pdf">Problems That Philippe Would Have Loved</a>, Paris 2014.

%e 0.1586682269720227755147804198163927392063737575530669779636652409544...

%p evalf[100]((3/7)*solve(11571875*x^5 -5363750*x^4 +628250*x^3 -97580*x^2 +5180*x -142=0, x)[1]); # _Muniru A Asiru_, Dec 21 2018

%o (PARI) (3/7)*solve(x=0, 1, 11571875*x^5 -5363750*x^4 +628250*x^3 -97580*x^2 +5180*x -142)

%Y Cf. A322631, A322632.

%K nonn,cons

%O 0,2

%A _Hugo Pfoertner_, Dec 21 2018