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 A182905 #22 Oct 16 2024 06:02:53
%S A182905 1,1,3,6,14,32,75,177,422,1013,2447,5942,14495,35501,87257,215144,
%T A182905 531970,1318726,3276644,8158736,20354413,50870857,127348839,319288920,
%U A182905 801657469,2015431885,5073224661,12785062080,32254748838,81457050078
%N A182905 Number of weighted lattice paths in F[n]. The members of F[n] are paths of weight n that start at (0,0), do not go below 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 A182905 The paths in F[n] need not end on the horizontal axis.
%C A182905 If f(z) is the generating function of the paths in F[n] (according to weight), and g(z) is the generating function of the those paths in F[n] that end on the horizontal axis, then f = g +z^2*gf. Also, g = (1 + zg)(1+z^2*g). Eliminating g from the above two equations, one obtains f(z).
%H A182905 M. Bona and A. Knopfmacher, <a href="http://dx.doi.org/10.1007/s00026-010-0060-7">On the probability that certain compositions have the same number of parts</a>, Ann. Comb., 14 (2010), 291-306.
%F A182905 G.f.: f(z)=2/[1-z-3z^2+sqrt((1+z+z^2)(1-3z+z^2))].
%F A182905 a(n) ~ sqrt(4935 + 2207*sqrt(5))* ((3 + sqrt(5))/2)^n / (sqrt(8*Pi)*n^(3/2)). - _Vaclav Kotesovec_, Mar 06 2016. Equivalently, a(n) ~ 5^(1/4) * phi^(2*n + 8) / (2*sqrt(Pi)*n^(3/2)), where phi = A001622 is the golden ratio. - _Vaclav Kotesovec_, Dec 06 2021
%F A182905 a(n) = Sum_{k=0..n} (k+1)*Sum_{i=0..n-k+1} C(i+1,n+1-i)*C(i+1,-i+n-k)/(i+1). - _Vladimir Kruchinin_, Jan 25 2019
%F A182905 D-finite with recurrence (n+2)*a(n) +(-4*n-5)*a(n-1) +(n-1)*a(n-2) +(4*n+5)*a(n-3) +(7*n-16)*a(n-4) +2*(n-1)*a(n-5) +2*(-n+4)*a(n-6)=0. - _R. J. Mathar_, Jul 24 2022
%e A182905 a(3)=6. Indeed, denoting by h (H) the (1,0)-step of weight 1 (2), and U=(1,1), D=(1,-1), we have hhh, hH, Hh, UD, hU, and Uh.
%p A182905 f := 2/(1-z-3*z^2+sqrt((1+z+z^2)*(1-3*z+z^2))): fser := series(f, z = 0, 32): seq(coeff(fser, z, n), n = 0 .. 29);
%t A182905 CoefficientList[Series[2/(1-x-3*x^2+Sqrt[(1+x+x^2)*(1-3*x+x^2)]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Mar 06 2016 *)
%o A182905 (Maxima)
%o A182905 a(n):=sum((k+1)*sum((binomial(i+1,n+1-i)*binomial(i+1,-i+n-k))/(i+1),i,0,n-k+1),k,0,n); /* _Vladimir Kruchinin_, Jan 25 2019 */
%K A182905 nonn
%O A182905 0,3
%A A182905 _Emeric Deutsch_, Dec 16 2010