A128720 Number of paths in the first quadrant from (0,0) to (n,0) using steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0).

1, 1, 3, 6, 16, 40, 109, 297, 836, 2377, 6869, 20042, 59071, 175453, 524881, 1579752, 4780656, 14536878, 44394980, 136107872, 418757483, 1292505121, 4001039563, 12418772656, 38641790001, 120510911885, 376628460529, 1179376013552, 3699860515924, 11626784875214

Offset: 0

Author

Emeric Deutsch, Mar 30 2007, revised Sep 03 2007

Keywords

• nonn

Comments

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).

Hankel transform is the (1,3) Somos-4 sequence A174168. - Paul Barry, Mar 10 2010

First column of the Riordan matrix A132276. - Emanuele Munarini, May 05 2011

Examples

			a(3)=6 because we have hhh, hH, Hh, hUD, UhD and UDh.
G.f. = 1 + x + 3*x^2 + 6*x^3 + 16*x^4 + 40*x^5 + 109*x^6 + 297*x^7 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000 (first 101 terms from Vincenzo Librandi)
• Paul Barry, Generalized Catalan Numbers, Hankel Transforms and Somos-4 Sequences , J. Int. Seq. 13 (2010) #10.7.2.
• Paul Barry, Invariant number triangles, eigentriangles and Somos-4 sequences, arXiv:1107.5490 [math.CO], 2011.
• Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019.
• Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
• Emeric Deutsch, Emanuele Munarini, and S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
• M. Dziemianczuk, Enumerations of plane trees with multiple edges and Raney lattice paths, Discrete Mathematics 337 (2014): 9-24.
• W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, A Pascal rhombus, Fibonacci Quarterly, 35 (1997), 318-328.
• P. Rajkovic, Paul Barry, and N. Savic, Number Sequences in an Integral Form with a Generalized Convolution Property and Somos-4 Hankel Determinants, Math. Balkanica, Vol. 26 (2012), Fasc. 1-2.

Crossrefs

Cf. A001006, A128719, A059397, A132276.

Programs

Maple

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);

Mathematica

Table[Sum[Binomial[2k,k]/(k+1)Sum[Binomial[n-j,2k]Binomial[n-j-2k,j],{j,0,n/2}],{k,0,n/2}],{n,0,12}] (* Emanuele Munarini, May 05 2011 *)

Maxima

makelist(sum(binomial(2*k,k)/(k+1)*sum(binomial(n-j,2*k)*binomial(n-j-2*k,j),j,0,n/2),k,0,n/2),n,0,12); /* Emanuele Munarini, May 05 2011 */

Formula

a(n) = Sum_{j=0..floor(n/2)} binomial(n-j, j)*m(n-2j), 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.

a(n) = a(n-1) + a(n-2) + Sum_{j=0..n-2} a(j)*a(n-2-j), a(0) = a(1) = 1.

G.f.: (1/(1-x-x^2))*c(x^2/(1-x-x^2)^2) = (1/(1-x^2))*m(x/(1-x^2)), c(x) the g.f. of A000108, m(x) the g.f. of A001006. - Paul Barry, Mar 18 2010

Let A(x) be the g.f., then B(x) = 1 + x*A(x) = 1 + 1*x + 1*x^2 + 3*x^3 + 6*x^4 + ... = 1/(1-z/(1-z/(1-z/(...)))) where z=x/(1+x-x^2) (continued fraction); more generally B(x)=C(x/(1+x-x^2)) where C(x) is the g.f. for the Catalan numbers (A000108). - Joerg Arndt, Mar 18 2011

a(n) = Sum_{k=0..floor(n/2)} (binomial(2*k,k)/(k+1))*Sum_{j=0..floor(n/2)} binomial(n-j, 2*k)*binomial(n-j-2*k, j). - Emanuele Munarini, May 05 2011

D-finite with recurrence: (n+2)*a(n) + (-2*n-1)*a(n-1) + 5*(-n+1)*a(n-2) + (2*n-5)*a(n-3) + (n-4)*a(n-4) = 0. - R. J. Mathar, Dec 03 2012

G.f.: (1 - x - x^2 - sqrt(1 - 2*x - 5*x^2 + 2*x^3 + x^4))/(2*x^2) = 1/Q(0), where Q(k) = 1 - x - x^2 - x^2/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Oct 04 2013

a(n) ~ sqrt(78+22*sqrt(13)) * ((3+sqrt(13))/2)^n / (4 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 13 2014