A059397 -id:A059397 - cp's OEIS frontend

A059398 Row sums of triangle in A059397.

1, 2, 6, 17, 51, 154, 473, 1464, 4568, 14332, 45187, 143024, 454217, 1446604, 4618576, 14777451, 47371177, 152110326, 489165277, 1575211177, 5078690936, 16392526502, 52963765321, 171282782902, 554393341371, 1795821017014

Offset: 0

N. J. A. Sloane, Jan 29 2001

Number of paths in the first quadrant from (0,0) to the line x=n, consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0) (in other words, left factors of the paths in A128720). Example: a(2)=6 because we have hh, H, UD, hU, Uh and UU. Row sums of triangle in A132276. - Emeric Deutsch, Sep 03 2007

Row sums of the Riordan matrix (g(x),x*g(x)), where g(x) = (1-x-x^2-sqrt(1-2*x-5*x^2+2*x^3+x^4))/(2*x^2) (A132276). - Emanuele Munarini, May 05 2011

G. C. Greubel, Table of n, a(n) for n = 0..1000
Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
W. Klostermeyer et al., A Pascal rhombus, Fibonacci Quarterly, 35 (1976), 318-328.

Cf. A128720, A132276.

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!(Sqrt((1+x-x^2)/(1-3*x-x^2))-1)/(2*x)) // G. C. Greubel, Jan 29 2018

g:=(1/2)*(sqrt((1+x-x^2)/(1-3*x-x^2))-1)/x: gser:=series(g,x=0,30): seq(coeff(gser,x,n),n=0..25); # Emeric Deutsch, Sep 03 2007

Table[Sum[Binomial[2k,k](-1)^(n-k+1)Sum[Binomial[i+k-1,i]Binomial[i,n-k-i+1],{i,0,n-k+1}],{k,0,n+1}]/2,{n,0,28}] (* Emanuele Munarini, May 05 2011 *)
With[{nn = 50}, CoefficientList[Series[(Sqrt[(1 + x - x^2)/(1 - 3*x - x^2)] - 1)/x/2, {x, 0, nn}], x]] (* G. C. Greubel, Jan 29 2018 *)

makelist(sum(binomial(2*k,k)*(-1)^(n-k+1)*sum(binomial(i+k-1,i)*binomial(i,n-k-i+1),i,0,n-k+1),k,0,n+1)/2,n,0,28); /* Emanuele Munarini, May 05 2011 */

x='x+O('x^30); Vec((sqrt((1+x-x^2)/(1-3*x-x^2))-1)/x/2) \\ G. C. Greubel, Jan 29 2018

G.f.: (sqrt((1+x-x^2)/(1-3*x-x^2))-1)/x/2. - Vladeta Jovovic, Jan 20 2004

a(n) = (1/2)*sum(binomial(2*k,k)*(-1)^(n-k+1)*sum(binomial(i+k-1,i)*binomial(i,n-k-i+1),i=0..n-k+1),k=0..n+1). - Emanuele Munarini, May 05 2011

More terms from Larry Reeves (larryr(AT)acm.org), Jan 31 2001

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).

Original entry on oeis.org

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

Emeric Deutsch, Mar 30 2007, revised Sep 03 2007

nonn

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

			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 + ...

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.

Cf. A001006, A128719, A059397, A132276.

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);

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 *)

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 */

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

A132276 Triangle read by rows: T(n,k) is the number of paths in the first quadrant from (0,0) to (n,k), consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0) (0<=k<=n).

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 6, 7, 3, 1, 16, 18, 12, 4, 1, 40, 53, 37, 18, 5, 1, 109, 148, 120, 64, 25, 6, 1, 297, 430, 369, 227, 100, 33, 7, 1, 836, 1244, 1146, 760, 385, 146, 42, 8, 1, 2377, 3656, 3519, 2518, 1391, 606, 203, 52, 9, 1, 6869, 10796, 10839, 8188, 4900, 2346, 903, 272

Offset: 0

Emeric Deutsch, Aug 16 2007, Sep 03 2007

Mirror image of A059397. - Emeric Deutsch, Aug 18 2007
Row sums yield A059398.
Riordan matrix (g(x),x*g(x)), where g(x) = (1-x-x^2-sqrt(1-2*x-5*x^2+2*x^3+x^4))/(2*x^2). - Emanuele Munarini, May 05 2011

			T(3,2) = 3 because we have UUh, UhU and hUU.
Triangle begins:
    1;
    1,   1;
    3,   2,   1;
    6,   7,   3,  1;
   16,  18,  12,  4,  1;
   40,  53,  37, 18,  5, 1;
  109, 148, 120, 64, 25, 6, 1;
  ...

Lin Yang and S.-L. Yang, The parametric Pascal rhombus. Fib. Q., 57:4 (2019), 337-346.

G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened
Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.
W. F. Klostermeyer, M. E. Mays, L. Soltes and G. Trapp, A Pascal rhombus, Fibonacci Quarterly, 35 (1997), 318-328.
Sheng-Liang Yang and Yuan-Yuan Gao, The Pascal rhombus and Riordan arrays, Fib. Q., 56:4 (2018), 337-347. See Fig. 1.

Cf. A059397, A128720 (the leading diagonal).

Cf. A059398.

g:=((1-z-z^2-sqrt((1+z-z^2)*(1-3*z-z^2)))*1/2)/z^2: G:=simplify(g/(1-t*z*g)): Gser:=simplify(series(G,z=0,13)): for n from 0 to 10 do P[n]:=sort(coeff(Gser, z,n)) end do: for n from 0 to 10 do seq(coeff(P[n], t, j), j = 0 .. n) end do; # yields sequence in triangular form

Flatten[Table[Sum[Binomial[2i+k,i]*(k+1)/(i+k+1)*Sum[Binomial[n-j,2i+k]*Binomial[n-k-2i-j,j],{j,0,n-k-2i}],{i,0,(n-k)/2}],{n,0,15},{k,0,n}]] (* Emanuele Munarini, May 05 2011 *)
c[x_] := (1 - Sqrt[1 - 4*x])/(2*x); g[z_] := c[z^2/(1 - z - z^2)^2]/(1 - z - z^2); G[t_, z_] := g[z]/(1 - t*z*g[z]); CoefficientList[ CoefficientList[Series[G[t, x], {x, 0, 49}, {t, 0, 49}], x], t]//Flatten (* G. C. Greubel, Dec 02 2017 *)

create_list(sum(binomial(2*i+k,i) * (k+1)/(i+k+1) * sum(binomial(n-j,2*i+k) * binomial(n-k-2*i-j,j),j,0,n-k-2*i), i,0,(n-k)/2), n,0,15, k,0,n); /* Emanuele Munarini, May 05 2011 */

T(n,k) = sum(i=0, (n-k)/2, (binomial(2*i+k,i)*(k+1)/(i+k+1))*sum(j=0, n-k-2*i, binomial(n-j,2*i+k)*binomial(n-k-2*i-j,j))) \\ G. C. Greubel, Nov 29 2017

T(n,0) = A128720(n).
G.f.: G(t,z) = g/(1-t*z*g), where g = 1 +z*g +z^2*g +z^2*g^2 or g = c(z^2/(1-z-z^2)^2)/(1-z-z^2), where c(z) = (1-sqrt(1-4*z))/(2*z) is the Catalan function.
T(n,k) = T(n-1,k-1) + T(n-1,k) + T(n-1,k+1) + T(n-2,k). - Emeric Deutsch, Aug 18 2007
Column k has g.f. z^k*g^(k+1), where g = 1 +z*g +z^2*g +z^2*g^2 = (1 -z-z^2 -sqrt((1+z-z^2)*(1-3*z-z^2)))/(2*z^2).
T(n,k) = Sum_{i=0..(n-k)/2} (binomial(2*i+k,i)*(k+1)/(i+k+1))*Sum_{j=0..(n-k-2*i)} binomial(n-j,2*i+k)*binomial(n-k-2*i-j,j). - Emanuele Munarini, May 05 2011 [corrected by Jason Yuen, Apr 08 2025]

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A059398 Row sums of triangle in A059397.

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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).

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A132276 Triangle read by rows: T(n,k) is the number of paths in the first quadrant from (0,0) to (n,k), consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0) (0<=k<=n).

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