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 A050168 #51 Oct 04 2024 00:25:44
%S A050168 1,2,3,5,9,16,30,55,105,196,378,714,1386,2640,5148,9867,19305,37180,
%T A050168 72930,140998,277134,537472,1058148,2057510,4056234,7904456,15600900,
%U A050168 30458900,60174900,117675360,232676280,455657715,901620585,1767883500
%N A050168 a(0) = 1; for n > 0, a(n) = binomial(n, floor(n/2)) + binomial(n-1, floor(n/2)).
%C A050168 a(n) = number of symmetric Dyck (n+1)-paths which either start UD or are prime, i.e., do not return to ground level until the terminal point. For example, a(2)=3 counts UUUDDD, UUDUDD, UDUDUD. - _David Callan_, Dec 09 2004
%C A050168 a(n) = number of symmetric Dyck (n+1)-paths that first return to ground level either right away or not until the very end, i.e., that remain Dyck paths when either the first two steps or the first and last steps are deleted. For example, a(2)=3 counts UUUDDD, UUDUDD, UDUDUD. - _David Callan_, Mar 02 2005
%C A050168 Hankel transform has g.f. (1-x(1+x)^2)/(1-x^2(1-x^2)). - _Paul Barry_, Sep 13 2007
%H A050168 Reinhard Zumkeller, <a href="/A050168/b050168.txt">Table of n, a(n) for n = 0..1000</a>
%H A050168 A. V. Sills and H. Wang, <a href="http://dx.doi.org/10.1016/j.dam.2012.03.002">On the maximal Wiener index and related questions</a>, Discrete Applied Mathematics, Volume 160, Issues 10-11, July 2012, Pages 1615-1623. - From _N. J. A. Sloane_, Sep 21 2012
%F A050168 Asymptotic to c*2^n/sqrt(n) where c = (3/4)*sqrt(2/Pi) = 0.598413... - _Benoit Cloitre_, Jan 13 2003
%F A050168 For n > 0: a(n) = A208976(n-1) + 1. -_Reinhard Zumkeller_, Mar 04 2012
%F A050168 Conjecture: (n+1)*a(n) + (n-3)*a(n-1) + 2*(-2*n+1)*a(n-2) + 4*(-n+3)*a(n-3) = 0. - _R. J. Mathar_, Nov 26 2012
%F A050168 G.f.: (1+x)/(2*x)*(sqrt((1+2*x)/(1-2*x))-1). - _Sergei N. Gladkovskii_, Jul 26 2013
%F A050168 G.f.: (1+x)/( W(0)*(1-2*x)*x) - (1+x)/(2*x), where W(k)= 1 + 1/(1 - 2*x/(2*x + (k+1)/(x*(2*k+1))/W(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Jul 26 2013
%t A050168 CoefficientList[(1+x)/(2x) (Sqrt[(1+2x)/(1-2x)]-1) + O[x]^34, x] (* _Jean-François Alcover_, Aug 04 2018, after _Sergei N. Gladkovskii_ *)
%o A050168 (Haskell)
%o A050168 a050168 n = a050168_list !! n
%o A050168 a050168_list = 1 : zipWith (+) a001405_list (tail a001405_list)
%o A050168 -- _Reinhard Zumkeller_, Mar 04 2012
%o A050168 (PARI) x='x+O('x^40); Vec((1+x)/(2*x)*(sqrt((1+2*x)/(1-2*x))-1)) \\ _G. C. Greubel_, Oct 26 2018
%o A050168 (Magma) m:=40; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1+x)/(2*x)*(Sqrt((1+2*x)/(1-2*x))-1))); // _G. C. Greubel_, Oct 26 2018
%Y A050168 Maximum element in n-th row of A029653 (generalized Pascal triangle).
%Y A050168 Cf. A001405.
%K A050168 nonn
%O A050168 0,2
%A A050168 _Clark Kimberling_