%I #51 Oct 04 2024 00:25:44
%S 1,2,3,5,9,16,30,55,105,196,378,714,1386,2640,5148,9867,19305,37180,
%T 72930,140998,277134,537472,1058148,2057510,4056234,7904456,15600900,
%U 30458900,60174900,117675360,232676280,455657715,901620585,1767883500
%N a(0) = 1; for n > 0, a(n) = binomial(n, floor(n/2)) + binomial(n-1, floor(n/2)).
%C 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 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 Hankel transform has g.f. (1-x(1+x)^2)/(1-x^2(1-x^2)). - _Paul Barry_, Sep 13 2007
%H Reinhard Zumkeller, <a href="/A050168/b050168.txt">Table of n, a(n) for n = 0..1000</a>
%H 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 Asymptotic to c*2^n/sqrt(n) where c = (3/4)*sqrt(2/Pi) = 0.598413... - _Benoit Cloitre_, Jan 13 2003
%F For n > 0: a(n) = A208976(n-1) + 1. -_Reinhard Zumkeller_, Mar 04 2012
%F 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 G.f.: (1+x)/(2*x)*(sqrt((1+2*x)/(1-2*x))-1). - _Sergei N. Gladkovskii_, Jul 26 2013
%F 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 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 (Haskell)
%o a050168 n = a050168_list !! n
%o a050168_list = 1 : zipWith (+) a001405_list (tail a001405_list)
%o -- _Reinhard Zumkeller_, Mar 04 2012
%o (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 (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 Maximum element in n-th row of A029653 (generalized Pascal triangle).
%Y Cf. A001405.
%K nonn
%O 0,2
%A _Clark Kimberling_