cp's OEIS Frontend

A137842 Number of paths from (0,0) if n is even, or from (2,1) if n is odd, to (3n,0) that stay in first quadrant (but may touch horizontal axis) and where each step is (2,1), (1,2) or (1,-1).

%I A137842 #7 Nov 22 2023 22:21:26
%S A137842 1,1,2,4,10,24,66,172,498,1360,4066,11444,34970,100520,312066,911068,
%T A137842 2862562,8457504,26824386,80006116,255680170,768464312,2471150402,
%U A137842 7474561164,24161357010,73473471344,238552980386,728745517972
%N A137842 Number of paths from (0,0) if n is even, or from (2,1) if n is odd, to (3n,0) that stay in first quadrant (but may touch horizontal axis) and where each step is (2,1), (1,2) or (1,-1).
%C A137842 Row sums of the inverse of the Riordan array (1/(1+x^2),x(1-x^2)/(1+x^2)).
%C A137842 a(n) is the maximum number of distinct sets that can be obtained as complete parenthesizations of “S_1 union S_2 intersect S_3 union S_4 intersect S_5 union ... S_{n+1}”, where the total of n union and intersection operations alternate, starting with a union, and S_1, S_2, ... , S_{n+1} are sets. - _Alexander Burstein_, Nov 22 2023
%F A137842 G.f.: (1+v^2)/(1-v), where v=2*sqrt(x^2+3)*sin(asin(x(x^2+18)/((x^2+3)^(3/2)))/3)/3-x/3; a(2n)=A027307(n); a(2n+1)=A032349(n+1).
%Y A137842 Cf. A084078. [From _R. J. Mathar_, Feb 28 2009]
%K A137842 easy,nonn
%O A137842 0,3
%A A137842 _Paul Barry_, Feb 13 2008