A376075 Number of North-East lattice paths from (0,0) to (n,n) that do not cross the diagonal y = x at any even point (2*k,2*k).

1, 2, 6, 14, 52, 140, 558, 1598, 6604, 19588, 82780, 251212, 1077992, 3324760, 14427422, 45039422, 197122524, 621205076, 2737289748, 8691699524, 38510822360, 123045322024, 547682980716, 1759017606220, 7859796084984, 25355507376808, 113670929821304

Offset: 0

Author

John Tyler Rascoe, Oct 08 2024

Keywords

• nonn
• easy

Examples

			The path NENNEENE does not cross y = x, so it is counted under a(4) = 52.
The path NENNENNEEEEN crosses y = x at points (1,1) and (5,5), so it is counted under a(6) = 558.
		

Crossrefs

Cf. A000108, A000984, A024492, A048990, A268400, A268429.

Programs

PARI

C(x) = {(1-sqrt(1-4*x))/(2*x)}
A(x) ={C(4*x)*C((x)*C(4*x))}
B(x) = {sqrt(C(4*x))}
D(x) = {1/sqrt(1-4*x)}
E_x(N) = {my(x='x+O('x^N));  Vec(D(x)-2*((C(x)-1)*((x*A(x^2))^2-B(x^2)^2+3*B(x^2)-2))/((2-B(x^2))*(2-C(x))))}
E_x(30)

Formula

G.f. D(x) - 2*((C(x) - 1)*((x*A(x^2))^2 - B(x^2)^2 + 3*B(x^2) - 2))/((2 - B(x^2))*(2 - C(x))), where A(x), B(x), C(x), and D(x) are the g.f.s for A024492, A048990, A000108, and A000984.