|
%I #14 Jul 24 2022 11:21:08
%S 2,14,106,862,7378,65550,599002,5594942,53181730,512784142,5003410762,
%T 49312114334,490192537586,4909102791694,49482525122490,
%U 501626536004734,5111038278845506,52312236295906830,537605889306476074
%N Number of returns to the x-axis 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, 368-370.
%F a(n) = Sum_{k=1..n} k*A108435(k).
%F a(n) = A032349(n+1) - A027307(n).
%F a(n) = 2 + (1/n)*Sum_{j=0..n-2} (3n-j)*2^(n-j)*binomial(n, j)*binomial(2n, n-j-1)/(n+j+2).
%F G.f.: A^2-A, where A=1+zA^2+zA^3 or, equivalently, A=(2/3)*sqrt((z+3)/z)*sin((1/3)*arcsin(sqrt(z)*(z+18)/(z+3)^(3/2)))-1/3 (the g.f. of A027307).
%F D-finite with recurrence 3*(n+1)*(2*n+1)*a(n) +3*(-34*n^2+18*n-1)*a(n-1) +(394*n^2-1197*n+908)*a(n-2) +2*(-4*n^2+51*n-113)*a(n-3) -2*(2*n-7)*(n-4)*a(n-4)=0. - _R. J. Mathar_, Jul 24 2022
%e a(2)=14 because there are 10 paths from (0,0) to (6,0) (see A027307): u(d)u(d), u(d)Ud(d), uud(d), uUdd(d), Ud(d)u(d), Ud(d)Ud(d), Udud(d), UdUdd(d), Uudd(d) and UUddd(d), the fourteen returns to the x-axis being shown between parentheses.
%p a:=n->2+(1/n)*sum((3*n-j)*2^(n-j)*binomial(n,j)*binomial(2*n,n-j-1)/(n+j+2),j=0..n-2): seq(a(n),n=1..21);
%Y Cf. A027307, A108435, A032349.
%K nonn
%O 1,1
%A _Emeric Deutsch_, Jun 04 2005
|
