%I #29 Dec 18 2019 09:09:18
%S 1,1,3,8,21,56,153,428,1222,3549,10454,31159,93801,284788,871007,
%T 2681018,8298932,25817395,80674901,253106836,796968055,2517706036,
%U 7977573202,25347126629,80738862084,257778971503,824798533932,2644335308021,8493626448823
%N Join 2n points on a line with n arcs above the line; form graph with the arcs as nodes, joining 2 nodes when the arcs cross. a(n) is the number of cases in which the graph is a path.
%C a(n+1) is the number of (finite) positive integer sequences b(1),...,b(k) with b(1) + Sum_{i=1..k-1} (1+max{b(i+1)-b(i), 0}) <= n. - Klaus Strassburger. [E.g., a(4)=8 since the integer sequences are 1; 2; 3; 1,1; 1,2; 2,1; 2,2; 1,1,1.]
%F G.f. (conjecture): 1 - G(0)/(1-x), where G(k)= 1 - 1/(1 - x/(1 - x/(1 - x/(1 - x/(x - 1/G(k+1)))))); (continued fraction). - _Sergei N. Gladkovskii_, Jul 12 2013
%F G.f. (conjecture): (2*x^3-x^2+2*x-1+sqrt(x^4+2*x^2-4*x+1))/(2*x^2-2*x). - _Michael D. Weiner_, Dec 17 2019
%K nonn
%O 1,3
%A _Martin Klazar_
%E More terms from Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de), Sep 24 2001
%E More terms from _Sean A. Irvine_, Apr 10 2018