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).

1, 1, 2, 4, 10, 24, 66, 172, 498, 1360, 4066, 11444, 34970, 100520, 312066, 911068, 2862562, 8457504, 26824386, 80006116, 255680170, 768464312, 2471150402, 7474561164, 24161357010, 73473471344, 238552980386, 728745517972

Offset: 0

Author

Paul Barry, Feb 13 2008

Keywords

• easy
• nonn

Comments

Row sums of the inverse of the Riordan array (1/(1+x^2),x(1-x^2)/(1+x^2)).

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

Crossrefs

Cf. A084078. [From R. J. Mathar, Feb 28 2009]

Formula

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).