This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A376075 #48 Oct 14 2024 02:56:32
%S A376075 1,2,6,14,52,140,558,1598,6604,19588,82780,251212,1077992,3324760,
%T A376075 14427422,45039422,197122524,621205076,2737289748,8691699524,
%U A376075 38510822360,123045322024,547682980716,1759017606220,7859796084984,25355507376808,113670929821304
%N 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).
%F A376075 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.
%e A376075 The path NENNEENE does not cross y = x, so it is counted under a(4) = 52.
%e A376075 The path NENNENNEEEEN crosses y = x at points (1,1) and (5,5), so it is counted under a(6) = 558.
%o A376075 (PARI)
%o A376075 C(x) = {(1-sqrt(1-4*x))/(2*x)}
%o A376075 A(x) ={C(4*x)*C((x)*C(4*x))}
%o A376075 B(x) = {sqrt(C(4*x))}
%o A376075 D(x) = {1/sqrt(1-4*x)}
%o A376075 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))))}
%o A376075 E_x(30)
%Y A376075 Cf. A000108, A000984, A024492, A048990, A268400, A268429.
%K A376075 nonn,easy
%O A376075 0,2
%A A376075 _John Tyler Rascoe_, Oct 08 2024