cp's OEIS Frontend

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).

%I A128720 #79 Jan 06 2025 13:48:03
%S A128720 1,1,3,6,16,40,109,297,836,2377,6869,20042,59071,175453,524881,
%T A128720 1579752,4780656,14536878,44394980,136107872,418757483,1292505121,
%U A128720 4001039563,12418772656,38641790001,120510911885,376628460529,1179376013552,3699860515924,11626784875214
%N 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).
%C A128720 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.
%C A128720 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).
%C A128720 Hankel transform is the (1,3) Somos-4 sequence A174168. - _Paul Barry_, Mar 10 2010
%C A128720 First column of the Riordan matrix A132276. - _Emanuele Munarini_, May 05 2011
%H A128720 G. C. Greubel, <a href="/A128720/b128720.txt">Table of n, a(n) for n = 0..1000</a> (first 101 terms from Vincenzo Librandi)
%H A128720 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Barry1/barry95r.html">Generalized Catalan Numbers, Hankel Transforms and Somos-4 Sequences </a>, J. Int. Seq. 13 (2010) #10.7.2.
%H A128720 Paul Barry, <a href="http://arxiv.org/abs/1107.5490">Invariant number triangles, eigentriangles and Somos-4 sequences</a>, arXiv:1107.5490 [math.CO], 2011.
%H A128720 Paul Barry, <a href="https://arxiv.org/abs/1910.00875">Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials</a>, arXiv:1910.00875 [math.CO], 2019.
%H A128720 Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Barry/barry601.html">On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
%H A128720 Emeric Deutsch, Emanuele Munarini, and S. Rinaldi, <a href="http://dx.doi.org/10.1016/j.jspi.2010.01.015">Skew Dyck paths</a>, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203.
%H A128720 M. Dziemianczuk, <a href="http://dx.doi.org/10.1016/j.disc.2014.07.024">Enumerations of plane trees with multiple edges and Raney lattice paths</a>, Discrete Mathematics 337 (2014): 9-24.
%H A128720 W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/35-4/klostermeyer.pdf">A Pascal rhombus</a>, Fibonacci Quarterly, 35 (1997), 318-328.
%H A128720 P. Rajkovic, Paul Barry, and N. Savic, <a href="http://www.math.bas.bg/infres/MathBalk/MB-26/MB-26-219-228.pdf">Number Sequences in an Integral Form with a Generalized Convolution Property and Somos-4 Hankel Determinants</a>, Math. Balkanica, Vol. 26 (2012), Fasc. 1-2.
%F A128720 a(n) = Sum_{j=0..floor(n/2)} binomial(n-j, j)*m(n-2j), where m(k)=A001006(k) are the Motzkin numbers.
%F A128720 G.f. = G satisfies z^2*G^2 - (1-z-z^2)*G + 1 = 0.
%F A128720 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.
%F A128720 a(n) = a(n-1) + a(n-2) + Sum_{j=0..n-2} a(j)*a(n-2-j), a(0) = a(1) = 1.
%F A128720 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
%F A128720 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
%F A128720 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
%F A128720 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
%F A128720 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
%F A128720 a(n) ~ sqrt(78+22*sqrt(13)) * ((3+sqrt(13))/2)^n / (4 * sqrt(Pi) * n^(3/2)). - _Vaclav Kotesovec_, Feb 13 2014
%e A128720 a(3)=6 because we have hhh, hH, Hh, hUD, UhD and UDh.
%e A128720 G.f. = 1 + x + 3*x^2 + 6*x^3 + 16*x^4 + 40*x^5 + 109*x^6 + 297*x^7 + ...
%p A128720 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);
%t A128720 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 *)
%o A128720 (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 */
%Y A128720 Cf. A001006, A128719, A059397, A132276.
%K A128720 nonn
%O A128720 0,3
%A A128720 _Emeric Deutsch_, Mar 30 2007, revised Sep 03 2007