%I
%S 3,31,311,3151,32299,334335,3488239,36627487,386618387,4098713631,
%T 43611791783,465496885231,4981942135611,53443871159551,
%U 574500093677535,6186886528903231,66735614131858723,720897596248427295
%N Number of d steps in all paths from (0,0) to (3n,0) that stay in the first quadrant (but may touch the horizontal axis), consisting of steps u=(2,1), U=(1,2), or d=(1,1).
%H Emeric Deutsch, <a href="http://www.jstor.org/stable/2589192">Problem 10658: Another Type of Lattice Path</a>, American Math. Monthly, 107, 2000, 368370.
%F a(n) = (1/n)*sum(k*binomial(n,2nk)*binomial(n+k,n1), k=n..2n).
%e a(1) = 3 because in the paths ud, Udd we have 3 d steps altogether.
%p a:=n>(1/n)*sum(k*binomial(n,2*nk)*binomial(n+k,n1),k=n..2*n): seq(a(n),n=1..22);
%Y Cf. A027307, A108429.
%K nonn
%O 1,1
%A _Emeric Deutsch_, Jun 03 2005
