A023431 -id:A023431 - cp's OEIS frontend

A090344 Number of Motzkin paths of length n with no level steps at odd level.

1, 1, 2, 3, 6, 11, 23, 47, 102, 221, 493, 1105, 2516, 5763, 13328, 30995, 72556, 170655, 403351, 957135, 2279948, 5449013, 13063596, 31406517, 75701508, 182902337, 442885683, 1074604289, 2612341856, 6361782007, 15518343597, 37912613631, 92758314874

Offset: 0

Emeric Deutsch, Jan 28 2004

nonn

a(n) = number of Motzkin paths of length n that avoid UF. Example: a(3) counts FFF, UDF, FUD but not UFD. - David Callan, Jul 15 2004

Also, number of 1-2 trees with n edges and with thinning limbs. A 1-2 tree is an ordered tree with vertices of outdegree at most 2. A rooted tree with thinning limbs is such that if a node has k children, all its children have at most k children. - Emeric Deutsch and Louis Shapiro, Nov 04 2006

			a(3)=3 because we have HHH, HUD and UDH, where U=(1,1), D=(1,-1) and H=(1,0).

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Andrei Asinowski, Cyril Banderier, and Valerie Roitner, Generating functions for lattice paths with several forbidden patterns, (2019).
Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009) 09.7.6.
Rui Duarte and António Guedes de Oliveira, Generating functions of lattice paths, Univ. do Porto (Portugal 2023).

Cf. A001006, A086622, A098474, A124344, A124497.

Cf. A014137, A023431, A144700.

[(&+[Binomial(n-k, k)*Catalan(k): k in [0..Floor(n/2)]]): n in [0..40]]; // G. C. Greubel, Jun 15 2022

C:=x->(1-sqrt(1-4*x))/2/x: G:=C(z^2/(1-z))/(1-z): Gser:=series(G,z=0,40): seq(coeff(Gser,z,n),n=0..36);
# second Maple program:
a:= proc(n) option remember; `if`(n<3, (n^2-n+2)/2,
     ((2*n+2)*a(n-1) -(4*n-6)*a(n-3) +(3*n-4)*a(n-2))/(n+2))
    end:
seq(a(n), n=0..40); # Alois P. Heinz, May 17 2013

Table[HypergeometricPFQ[{1/2, (1-n)/2, -n/2}, {2, -n}, -16], {n, 0, 40}] (* Jean-François Alcover, Feb 20 2015, after Paul Barry *)

{a(n)=local(A=1+x);for(i=1,n,A=1/(1-x+x*O(x^n))+x^2*A^2+x*O(x^n));polcoeff(A,n)} \\ Paul D. Hanna, Jun 24 2012

[sum(binomial(n-k,k)*catalan_number(k) for k in (0..(n//2))) for n in (0..40)] # G. C. Greubel, Jun 15 2022

G.f.: (1-x-sqrt(1-2*x-3*x^2+4*x^3))/(2*x^2*(1-x)).
G.f. satisfies: A(x) = 1/(1-x) + x^2*A(x)^2. - Paul D. Hanna, Jun 24 2012
D-finite with recurrence (n+2)*a(n) = 2*(n+1)*a(n-1) + (3*n-4)*a(n-2) - 2*(2*n-3)*a(n-3). - Vladeta Jovovic, Sep 11 2004
a(n) = Sum_{k=0..floor(n/2)} binomial(n-k, k)*binomial(2*k, k)/(k+1). - Paul Barry, Nov 13 2004
a(n) = 1 + Sum_{k=1..n-1} a(k-1)*a(n-k-1). - Henry Bottomley, Feb 22 2005
G.f.: 1/(1-x-x^2/(1-x^2/(1-x-x^2/(1-x^2/(1-x-x^2/(1-x^2/(1-... (continued fraction). - Paul Barry, Apr 08 2009
With M = an infinite tridiagonal matrix with all 1's in the super and subdiagonals and [1,0,1,0,1,0,...] in the main diagonal and V = vector [1,0,0,0,...] with the rest zeros, the sequence starting with offset 1 = leftmost column iterates of M*V. - Gary W. Adamson, Jun 08 2011
Recurrence (an alternative): (n+2)*a(n) = 3*(n+1)*a(n-1) + (n-4)*a(n-2) - (7*n-13)*a(n-3) + 2*(2*n-5)*a(n-4), n>=4. - Fung Lam, Apr 01 2014
Asymptotics: a(n) ~ (8/(sqrt(17)-1))^n*( 1/17^(1/4) + 17^(1/4) )*17 /(16*sqrt(Pi*n^3)). - Fung Lam, Apr 01 2014
a(n) = 2*A026569(n) + A026569(n+1)/2 - A026569(n+2)/2. - Mark van Hoeij, Nov 29 2024

A052723 Expansion of e.g.f. (1 - x - sqrt(1-2x+x^2-4x^3))/(2*x).

Original entry on oeis.org

0, 0, 2, 6, 24, 240, 2880, 35280, 524160, 9434880, 188697600, 4151347200, 101548339200, 2727435110400, 79332244992000, 2488504322304000, 83879464660992000, 3021209014247424000, 115754916599562240000

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

G. C. Greubel, Table of n, a(n) for n = 0..350
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 679

Cf. A052711, A052712, A052713, A052714, A052715, A052716, A052717, A052718, A052719, A052720, A052721, A052722.

Cf. A023431.

spec := [S,{B=Prod(S,S),C=Union(B,S,Z),S=Prod(Z,C)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
seq(n!*add(binomial(n-2-k,2*k)*binomial(2*k,k)/(k+1), k=0..floor((n-2)/3)), n=0..18);  # Mark van Hoeij, May 12 2013

With[{nn=20},CoefficientList[Series[(1-x-Sqrt[1-2x+x^2-4x^3])/(2x),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jan 19 2017 *)
a[n_]:= a[n]= n!*Sum[Binomial[n-k-2,2*k]*CatalanNumber[k], {k,0,Floor[(n-2)/2]}];
Table[a[n], {n,0,30}] (* G. C. Greubel, May 28 2022 *)

def A052723(n): return factorial(n)*sum( binomial(n-k-2, 2*k)*catalan_number(k) for k in (0..(n-2)//2) )
[A052723(n) for n in (0..30)] # G. C. Greubel, May 28 2022

D-finite with recurrence: a(0) = a(1) = 0, a(2) = 2, a(3) = 6, a(4) = 24, (n+4)*a(n+3) = (15 + 11*n + 2*n^2)*a(n+2) - (6 + 11*n + 6*n^2 + n^3)*a(n+1) - (12 - 2*n - 32*n^2 - 22*n^2 - 4*n^4)*a(n).

a(n) = n!*A023431(n-2). - R. J. Mathar, Oct 18 2013

A025246 a(n) = a(1)a(n-1) + a(2)a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 1, 0, 1, 1.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 4, 7, 13, 26, 52, 104, 212, 438, 910, 1903, 4009, 8494, 18080, 38656, 82988, 178802, 386490, 837928, 1821664, 3970282, 8673258, 18987930, 41652382, 91539466, 201525238, 444379907, 981384125, 2170416738, 4806513660, 10657780276

Offset: 1

Clark Kimberling

nonn

Essentially the same as A023431.

G. C. Greubel, Table of n, a(n) for n = 1..1000
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 666

Cf. A000108, A023431.

[n le 2 select 2-n else (&+[Binomial(n-k-3, 2*k)*Catalan(k): k in [0..Floor((n-3)/3)]]): n in [1..45]]; // G. C. Greubel, Jun 15 2022

a := n -> ifelse(n < 3, 0^(n - 1),
   hypergeom([(3 - n)/3, (4 - n)/3, (5 - n)/3], [2, -n + 3], 27)):
seq(simplify(a(n)), n = 1..32); # Peter Luschny, Jun 15 2022

a[n_]:= a[n]= If[n<4, 1-Boole[n==2], Sum[a[j]*a[n-j], {j,n-3}]];
Table[a[n], {n, 45}] (* G. C. Greubel, Jun 15 2022 *)

a(n)=polcoeff((1+x-sqrt(1-2*x+x^2-4*x^3+x*O(x^n)))/2,n)

[bool(n==1) + sum(binomial(n-k-3,2*k)*catalan_number(k) for k in (0..((n-3)//3))) for n in (1..45)] # G. C. Greubel, Jun 15 2022

a(n) = A023431(n-3).
G.f.: (1+x-sqrt(1-2*x+x^2-4*x^3))/2. - Michael Somos, Jun 08 2000
n*a(n) = (2*n-3)*a(n-1) -(n-3)*a(n-2) +2*(2*n-9)*a(n-3). - R. J. Mathar, Feb 25 2015
a(n) = hypergeom([(3 - n)/3, (4 - n)/3, (5 - n)/3], [2, 3 - n], 27) for n >= 3. - Peter Luschny, Jun 15 2022

A052702 Expansion of (1/2)(1/x^2 - 1/x)(1-x-sqrt(1-2x+x^2-4x^3)) - x.

Original entry on oeis.org

0, 0, 0, 0, 1, 2, 3, 6, 13, 26, 52, 108, 226, 472, 993, 2106, 4485, 9586, 20576, 44332, 95814, 207688, 451438, 983736, 2148618, 4702976, 10314672, 22664452, 49887084, 109985772, 242854669, 537004218, 1189032613, 2636096922, 5851266616, 13002628132, 28925389870, 64412505472, 143576017410

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

From Paul Barry, May 24 2009: (Start)
Hankel transform of A052702 is A160705. Hankel transform of A052702(n+1) is A160706.
Hankel transform of A052702(n+2) is -A131531(n+1). Hankel transform of A052702(n+3) is A160706(n+5).
Hankel transform of A052702(n+4) is A160705(n+5). (End)
For n > 1, number of Dyck (n-1)-paths with each descent length one greater or one less than the preceding ascent length. - David Scambler, May 11 2012

Ricardo Gómez Aíza, Trees with flowers: A catalog of integer partition and integer composition trees with their asymptotic analysis, arXiv:2402.16111 [math.CO], 2024. See pp. 10, 19-21.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 654

Cf. A023431.

spec := [S,{B=Prod(C,Z),S=Prod(B,B),C=Union(S,B,Z)},unlabeled]: seq(combstruct[count](spec, size=n), n=0..20);

a[n_] := Sum[Binomial[n-k-2, 2k-1] CatalanNumber[k], {k, 0, n-2}];
Table[a[n], {n, 0, 40}] (* Jean-François Alcover, Oct 11 2022, after Paul Barry *)

x='x+O('x^66);
s='a0+(1-2*x+x^2-2*x^3-(1-x)*sqrt(1-2*x+x^2-4*x^3))/(2*x^2);
v=Vec(s);  v[1]-='a0;  v
/* Joerg Arndt, May 11 2012 */

Recurrence: {a(1)=0, a(2)=0, a(4)=1, a(3)=0, a(6)=3, a(7)=6, a(5)=2, (-2+4*n)*a(n)+(-7-5*n)*a(n+1)+(8+3*n)*a(n+2)+(-13-3*n)*a(n+3)+(n+6)*a(n+4)}.
From Paul Barry, May 24 2009: (Start)
G.f.: (1-2*x+x^2-2*x^3-(1-x)*sqrt(1-2*x+x^2-4*x^3))/(2*x^2).
a(n+1) = Sum_{k=0..n-1} C(n-k-1,2k-1)*A000108(k). (End)
a(n) = A023431(n-1)-A023431(n-2). - R. J. Mathar, Jan 13 2025

More terms from Joerg Arndt, May 11 2012

A357308 a(0) = a(1) = 0, a(2) = 1; a(n) = a(n-1) + Sum_{k=0..n-3} a(k) * a(n-k-3).

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 2, 4, 7, 11, 16, 24, 39, 67, 116, 196, 324, 534, 892, 1516, 2601, 4463, 7630, 13022, 22276, 38286, 66084, 114328, 197929, 342783, 594218, 1031794, 1794944, 3127450, 5455272, 9523812, 16640542, 29102938, 50951070, 89289998, 156616648, 274923328, 482945930, 848972814

Offset: 0

Ilya Gutkovskiy, Sep 23 2022

nonn

Cf. A006318, A023426, A023431, A025246, A346503, A346504, A357307.

a[0] = a[1] = 0; a[2] = 1; a[n_] := a[n] = a[n - 1] + Sum[a[k] a[n - k - 3], {k, 0, n - 3}]; Table[a[n], {n, 0, 44}]
nmax = 44; A[] = 0; Do[A[x] = x^2 (1 + x A[x]^2)/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]

G.f. A(x) satisfies: A(x) = x^2 * (1 + x * A(x)^2) / (1 - x).

A144700 Generalized (3,-1) Catalan numbers.

Original entry on oeis.org

1, 1, 1, 1, 2, 5, 11, 21, 38, 71, 141, 289, 591, 1195, 2410, 4897, 10051, 20763, 42996, 89139, 185170, 385809, 806349, 1689573, 3547152, 7459715, 15714655, 33161821, 70095642, 148388521, 314562189, 667682057, 1418942341

Offset: 0

Paul Barry, Sep 19 2008

Number of lattice paths in the first quadrant from (0,0) to (n,0) using only steps H=(1,0), U=(1,1) and D=(3,-1). Hankel transform has g.f. (1-x^3)/(1+x^4) (A132380 (n+3)).

G. C. Greubel, Table of n, a(n) for n = 0..1000
S. B. Ekhad and M. Yang, Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences, (2017).

Cf. A000108, A014137, A023431, A090344, A132380.

[(&+[Binomial(n-k,3*k)*Catalan(k): k in [0..Floor(n/4)]]): n in [0..40]]; // G. C. Greubel, Jun 15 2022

b[n_, m_]:=a[n, m]=Sum[Binomial[n-k,m*k]*CatalanNumber[k], {k,0,Floor[n/(m+1)]}];
A144700[n_]:= b[n,3]; (* A014137 (m=0), A090344 (m=1), A023431 (m=2) *)
Table[A144700[n], {n, 0, 40}] (* G. C. Greubel, Jun 15 2022 *)

[sum(binomial(n-k,3*k)*catalan_number(k) for k in (0..(n//4))) for n in (0..40)] # G. C. Greubel, Jun 15 2022

G.f.: (1/(1-x)) * c(x^4/(1-x)^3), where c(x) is the g.f. of A000108.
a(n) = Sum_{k=0..floor(n/4)} binomial(n-k, 3*k)*A000108(k).
(n+4)*a(n) = 2*(2*n+5)*a(n-1) - 6*(n+1)*a(n-2) + 2*(2*n-1)*a(n-3) +3*(n-2)*a(n-4) - 2*(2*n-7)*a(n-5). - R. J. Mathar, Nov 16 2011
a(n) = b(n, 3), where b(n, m) = Sum_{k=0..floor(n/(m+1))} binomial(n-k, m*k)*A000108(k). - G. C. Greubel, Jun 15 2022

A346503 G.f. A(x) satisfies A(x) = 1 + x^3 * A(x)^2 / (1 - x).

Original entry on oeis.org

1, 0, 0, 1, 1, 1, 3, 5, 7, 14, 26, 43, 79, 148, 264, 483, 903, 1664, 3080, 5771, 10795, 20209, 38059, 71799, 135569, 256762, 487310, 925981, 1762841, 3361897, 6419595, 12275301, 23505143, 45061424, 86485016, 166176499, 319630115, 615387675, 1185940209, 2287527119, 4416083429

Offset: 0

Ilya Gutkovskiy, Jul 21 2021

nonn

Cf. A002212, A023426, A023431, A090345, A216604 (partial sums), A346504, A366588, A376490.

nmax = 40; A[] = 0; Do[A[x] = 1 + x^3 A[x]^2/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[1] = a[2] = 0; a[n_] := a[n] = a[n - 1] + Sum[a[k] a[n - k - 3], {k, 0, n - 3}]; Table[a[n], {n, 0, 40}]

a(0) = 1, a(1) = a(2) = 0; a(n) = a(n-1) + Sum_{k=0..n-3} a(k) * a(n-k-3).
a(n) ~ 2^(n+1) / (sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Jul 30 2021
From Seiichi Manyama, Sep 26 2024: (Start)
G.f.: 2/(1 + sqrt(1 - 4*x^3/(1 - x))).
a(n) = Sum_{k=0..floor(n/3)} binomial(2*k,k) * binomial(n-2*k-1,n-3*k) / (k+1). (End)

A346504 G.f. A(x) satisfies: A(x) = 1 + x + x^3 * A(x)^2 / (1 - x).

Original entry on oeis.org

1, 1, 0, 1, 3, 4, 6, 14, 28, 49, 95, 196, 386, 754, 1524, 3102, 6258, 12700, 26032, 53440, 109772, 226457, 468863, 972300, 2020274, 4208530, 8784556, 18365322, 38461110, 80682740, 169501696, 356579216, 751138916, 1584281062, 3345404514, 7072055268, 14965933024, 31702754496

Offset: 0

Ilya Gutkovskiy, Jul 21 2021

nonn

Cf. A023426, A023431, A025243, A090345, A248100, A257290, A346503.

nmax = 37; A[] = 0; Do[A[x] = 1 + x + x^3 A[x]^2/(1 - x) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
a[0] = a[1] = 1; a[2] = 0; a[n_] := a[n] = a[n - 1] + Sum[a[k] a[n - k - 3], {k, 0, n - 3}]; Table[a[n], {n, 0, 37}]
CoefficientList[Series[(1 - x)*(1 - Sqrt[(1 - x - 4*x^3 - 4*x^4)/(1 - x)]) / (2*x^3), {x, 0, 40}], x] (* Vaclav Kotesovec, Sep 27 2023 *)

a(0) = a(1) = 1, a(2) = 0; a(n) = a(n-1) + Sum_{k=0..n-3} a(k) * a(n-k-3).

G.f.: (1-x)*(1 - sqrt((1 - x - 4*x^3 - 4*x^4)/(1-x))) / (2*x^3). - Vaclav Kotesovec, Sep 27 2023

A349047 G.f. A(x) satisfies: A(x) = 1 / (1 - x + x^3 * A(x)).

Original entry on oeis.org

1, 1, 1, 0, -2, -5, -7, -4, 10, 38, 70, 68, -40, -329, -767, -1012, -214, 2842, 8642, 14332, 10136, -21622, -96578, -196412, -213080, 96264, 1037344, 2608788, 3698996, 1121127, -10234567, -33425980, -58537486, -45735382, 83471346, 408899204, 871127040

Offset: 0

Ilya Gutkovskiy, Nov 06 2021

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A000108, A007440, A023431, A100223, A214649, A343773, A349048.

nmax = 36; A[] = 0; Do[A[x] = 1/(1 - x + x^3 A[x]) + O[x]^(nmax + 1) // Normal,nmax + 1]; CoefficientList[A[x], x]
a[0] = 1; a[n_] := a[n] = a[n - 1] - Sum[a[k] a[n - k - 3], {k, 0, n - 3}]; Table[a[n], {n, 0, 36}]
Table[Sum[(-1)^k Binomial[n - k, 2 k] CatalanNumber[k], {k, 0, Floor[n/3]}], {n, 0, 36}]

G.f.: (-1 + x + sqrt((1 - x)^2 + 4*x^3)) / (2*x^3).
a(0) = 1; a(n) = a(n-1) - Sum_{k=0..n-3} a(k) * a(n-k-3).
a(n) = Sum_{k=0..floor(n/3)} (-1)^k * binomial(n-k,2*k) * Catalan(k).
a(n) = F([(1-n)/3, (2-n)/3, -n/3], [2, -n], -27), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 07 2021

A276066 Triangle read by rows: T(n,k) is the number of bargraphs of semiperimeter n having a total of k double rises and double falls (n>=2,k>=0). A double rise (fall) in a bargraph is any pair of adjacent up (down) steps.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 1, 0, 1, 2, 4, 1, 4, 1, 0, 1, 4, 6, 8, 8, 1, 6, 1, 0, 1, 7, 14, 22, 12, 19, 12, 1, 8, 1, 0, 1, 13, 34, 43, 48, 55, 18, 35, 16, 1, 10, 1, 0, 1, 26, 72, 105, 148, 109, 116, 103, 24, 56, 20, 1, 12, 1, 0, 1, 52, 154, 276, 344, 347, 398, 205, 232, 166, 30, 82, 24, 1, 14, 1, 0, 1

Offset: 2

Emeric Deutsch and Sergi Elizalde, Aug 25 2016

Number of entries in row n is 2n-3.
Sum of entries in row n = A082582(n).
T(n,0) = A023431(n-2) = A025246(n+1).
Sum(k*T(n,k),k>=0) = 2*A273714(n).

			Row 4 is 1,2,1,0,1 because the 5 (=A082582(4)) bargraphs of semiperimeter 4 correspond to the compositions [1,1,1], [1,2], [2,1], [2,2], [3] and the corresponding drawings show that they have a total of  0, 1, 1, 2, 4 double rises and double falls, respectively.
Triangle starts:
1;
1,0,1;
1,2,1,0,1;
2,4,1,4,1,0,1;
4,6,8,8,1,6,1,0,1.

Alois P. Heinz, Rows n = 2..140, flattened
M. Bousquet-Mélou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.
Emeric Deutsch, S Elizalde, Statistics on bargraphs viewed as cornerless Motzkin paths, arXiv preprint arXiv:1609.00088, 2016

Cf. A023431, A025246, A082582, A273714.

eq := z*G^2-(1-z-t^2*z-2*t*z^2+t^2*z^2)*G+z^2 = 0: G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 22)): for n from 2 to 20 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 2 to 20 do seq(coeff(P[n], t, j), j = 0 .. 2*n-4) end do; # yields sequence in triangular form.
# second Maple program:
b:= proc(n, y, t) option remember; expand(`if`(n=0, (1-t)*
      z^(y-1), `if`(t<0, 0, b(n-1, y+1, 1)*`if`(t=1, z, 1))+
     `if`(t>0 or y<2, 0, b(n, y-1, -1)*`if`(t=-1, z, 1))+
     `if`(y<1, 0, b(n-1, y, 0))))
    end:
T:= n->(p->seq(coeff(p, z, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=2..12);  # Alois P. Heinz, Aug 25 2016

b[n_, y_, t_] := b[n, y, t] = Expand[If[n == 0, (1 - t)*z^(y - 1), If[t < 0, 0, b[n - 1, y + 1, 1]*If[t == 1, z, 1]] + If[t > 0 || y < 2, 0, b[n, y - 1, -1]*If[t == -1, z, 1]] + If[y < 1, 0, b[n - 1, y, 0]]]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 0, Exponent[p, z]}]][b[n, 0, 0]]; Table[T[n], {n, 2, 12}] // Flatten (* Jean-François Alcover, Dec 02 2016 after Alois P. Heinz *)

G.f.: G = G(t,z) satisfies zG^2 - (1-z - t^2*z - 2tz^2+t^2*z^2)G + z^2 = 0.

The g.f. B(t,s,z) of bargraphs, where t(s) marks double rises (falls) and z marks semiperimeter, satisfies zB^2 - (1-(1+ts)z +(ts- t-s)z^2)B + z^2 = 0.

cp's OEIS Frontend

A090344 Number of Motzkin paths of length n with no level steps at odd level.

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A052723 Expansion of e.g.f. (1 - x - sqrt(1-2*x+x^2-4*x^3))/(2*x).

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A025246 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 1, 0, 1, 1.

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A052702 Expansion of (1/2)*(1/x^2 - 1/x)*(1-x-sqrt(1-2*x+x^2-4*x^3)) - x.

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A357308 a(0) = a(1) = 0, a(2) = 1; a(n) = a(n-1) + Sum_{k=0..n-3} a(k) * a(n-k-3).

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A144700 Generalized (3,-1) Catalan numbers.

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A346503 G.f. A(x) satisfies A(x) = 1 + x^3 * A(x)^2 / (1 - x).

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A052723 Expansion of e.g.f. (1 - x - sqrt(1-2x+x^2-4x^3))/(2*x).

A025246 a(n) = a(1)a(n-1) + a(2)a(n-2) + ... + a(n-3)*a(3) for n >= 4, with initial terms 1, 0, 1, 1.

A052702 Expansion of (1/2)(1/x^2 - 1/x)(1-x-sqrt(1-2x+x^2-4x^3)) - x.