0,3

Points of two kinds are placed on a line: light points having weight 1 and heavy points having weight 2. Number of configurations of points of total weight n, with some of the light points being paired off by nonintersecting arcs.

Number of skew Dyck paths of semilength n having no UUU's. A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of a path is defined to be the number of its steps. Example: a(3)=6 because we have UDUDUD, UDUUDD, UDUUDL, UUDDUD, UUDUDD and UUDUDL. a(n)=A128719(n,0). a(n)=A059397(n,n). a(n)=A132276(n,0).

E. Deutsch, E. Munarini and S. Rinaldi, Skew Dyck paths (in preparation).

W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, A Pascal rhombus, Fibonacci Quarterly, 35 (1997), 318-328.

a(n)=Sum(binom(n-j, j)*m(n-2j), j=0..floor(n/2)), where m(k)=A001006(k) are the Motzkin numbers. G.f.=G satisfies z^2*G^2-(1-z-z^2)G+1=0. G.f.=c(z^2/(1-z-z^2)^2)/(1-z-z^2), where c(z)=[1-sqrt(1-4z)]/(2z) is the Catalan function. Rec. rel.: a(n)=a(n-1)+a(n-2)+Sum(a(j)a(n-2-j), j=0..n-2); a(0)=a(1)=1.

a(3)=6 because we have hhh, hH, Hh, hUD, UhD and UDh.

a[0]:=1: a[1]:=1: for n from 2 to 30 do a[n]:=a[n-1]+a[n-2]+add(a[j]*a[n-2-j], j=0..n-2) end do: seq(a[n], n=0..30); G:=((1-z-z^2-sqrt((1+z-z^2)*(1-3*z-z^2)))*1/2)/z^2: Gser:=series(G, z=0, 33): seq(coeff(Gser, z, n), n=0..30);

Cf. A001006, A128719, A059397, A132276.

Sequence in context: A046211 A018022 A166536 this_sequence A096745 A027088 A027102

Adjacent sequences: A128717 A128718 A128719 this_sequence A128721 A128722 A128723

nonn

Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 30 2007, revised Sep 03 2007