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 A182897 #10 Jul 24 2022 13:53:29
%S A182897 0,0,0,1,3,9,27,76,211,580,1578,4267,11484,30789,82301,219465,584060,
%T A182897 1551770,4117061,10910049,28881387,76387179,201875129,533145603,
%U A182897 1407161007,3711981168,9787157469,25793933410,67952779665,178954077522
%N A182897 Number of (1,-1)-returns to the horizontal axis in all weighted lattice paths in L_n. The members of L_n are paths of weight n that start at (0,0) , end on the horizontal axis and whose steps are of the following four kinds: an (1,0)-step with weight 1, an (1,0)-step with weight 2, a (1,1)-step with weight 2, and a (1,-1)-step with weight 1. The weight of a path is the sum of the weights of its steps.
%C A182897 a(n)=Sum(k*A182896(n,k), k>=0).
%D A182897 M. Bona and A. Knopfmacher, On the probability that certain compositions have the same number of parts, Ann. Comb., 14 (2010), 291-306.
%D A182897 E. Munarini, N. Zagaglia Salvi, On the rank polynomial of the lattice of order ideals of fences and crowns, Discrete Mathematics 259 (2002), 163-177.
%F A182897 G.f.: G(z)=z^3*c/[(1+z+z^2)(1-3z+z^2)], where c satisfies c = 1+zc+z^2*c+z^3*c^2.
%F A182897 a(n) ~ ((1 + sqrt(5))/2)^(2*n+1) / (4*sqrt(5)). - _Vaclav Kotesovec_, Mar 06 2016
%F A182897 D-finite with recurrence n*a(n) +(-4*n+3)*a(n-1) +(2*n-3)*a(n-2) +11*(n-3)*a(n-4) +(2*n-9)*a(n-6) +(-4*n+21)*a(n-7) +(n-6)*a(n-8)=0. - _R. J. Mathar_, Jul 24 2022
%e A182897 a(3)=1 because, denoting by h (H) the (1,0)-step of weight 1 (2), and u=(1,1), d=(1,-1), the five paths of weight 3 are ud, du, hH, Hh, and hhh; they contain 1+0+0+0+0=1 (1,-1)-return to the horizontal axis.
%p A182897 eq := c = 1+z*c+z^2*c+z^3*c^2: c := RootOf(eq, c): G := z^3*c/((1+z+z^2)*(1-3*z+z^2)): Gser := series(G, z = 0, 32): seq(coeff(Gser, z, n), n = 0 .. 29);
%t A182897 CoefficientList[Series[x^3*(1 - x - x^2 - Sqrt[1+x^4-2*x^3-x^2-2*x]) / (2*x^3*(1+x+x^2)*(1-3*x+x^2)), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 06 2016 *)
%Y A182897 A182896
%K A182897 nonn
%O A182897 0,5
%A A182897 _Emeric Deutsch_, Dec 13 2010