cp's OEIS Frontend

A268429 Number of North-East lattice paths from (0,0) to (n,n) that bounce off the diagonal y = x to the right exactly once.

%I A268429 #17 Aug 28 2024 00:55:03
%S A268429 1,4,16,62,238,910,3475,13270,50707,193948,742659,2847126,10928009,
%T A268429 41993692,161555008,622201838,2398811962,9257512318,35760612784,
%U A268429 138263710226,535038428936,2072130742074,8031333322206,31151602276002,120915026597458,469648731423190,1825348333058230,7098811400187410,27623655321103718
%N A268429 Number of North-East lattice paths from (0,0) to (n,n) that bounce off the diagonal y = x to the right exactly once.
%C A268429 This sequence is related to paired pattern P_2 in Section 3.2 in Pan and Remmel's link.
%C A268429 By symmetry, a(n) is also the number of North-East lattice paths from (0,0) to (n,n) that bounce off the diagonal y = x to the left exactly once.
%H A268429 John Tyler Rascoe, <a href="/A268429/b268429.txt">Table of n, a(n) for n = 2..100</a>
%H A268429 Ran Pan and Jeffrey B. Remmel, <a href="http://arxiv.org/abs/1601.07988">Paired patterns in lattice paths</a>, arXiv:1601.07988 [math.CO], 2016.
%F A268429 G.f.: (-1 + f(x) + 2*x)^2/(1 - f(x) + (-5 + f(x))*x)^2, where f(x) = sqrt(1 - 4*x).
%t A268429 (1 - 4x + x^2 - 2x^3 - Sqrt[1-4x](1 - 2x - 3x^2))/(2(-1 + x(4+x))^2) + O[x]^31 // CoefficientList[#, x]& // Drop[#, 2]& (* _Jean-François Alcover_, Dec 15 2018 *)
%o A268429 (PARI)
%o A268429 A_x(N) = {my(x='x+O('x^N), f=sqrt(1-4*x)); Vec((-1+f+2*x)^2/(1-f+(-5+f)*x)^2)}
%o A268429 A_x(31) \\ _John Tyler Rascoe_, Aug 26 2024
%Y A268429 Cf. A000108, A268407, A268429.
%K A268429 nonn
%O A268429 2,2
%A A268429 _Ran Pan_, Feb 04 2016