A007226 -id:A007226 - cp's OEIS frontend

A334648 a(n) is the total number of down steps between the second and third up steps in all 3_1-Dyck paths of length 4*n.

0, 0, 34, 132, 722, 4638, 32416, 238956, 1827918, 14370595, 115384756, 942115942, 7798224226, 65286060253, 551838621972, 4702955036640, 40366238473530, 348631520142879, 3027590307082804, 26420699531880832, 231571468023697960, 2037650653547067005

Offset: 0

Benjamin Hackl, May 12 2020

A 3_1-Dyck path is a lattice path with steps (1, 3), (1, -1) that starts and ends at y = 0 and stays above the line y = -1.

For n = 2, there is no 3rd up step, a(2) = 34 enumerates the total number of down steps between the 2nd up step and the end of the path.

			For n = 2, the 3_1-Dyck paths are DUDDDUDD, DUDDUDDD, DUDUDDDD, DUUDDDDD, UDDDDUDD, UDDDUDDD, UDDUDDDD, UDUDDDDD, UUDDDDDD. In total, there are a(2) = 2 + 3 + 4 + 5 + 2 + 3 + 4 + 5 + 6 = 34 down steps between the 2nd up step and the end of the path.

Andrei Asinowski, Benjamin Hackl, Sarah J. Selkirk, Down-step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

Cf. A002293, A007226, A007228, A334645, A334646, A334647, A334649, A334680, A334682, A334785.

a[0] = a[1] = 0; a[n_] := Binomial[4*n + 1, n]/(4*n + 1) + 6 * Sum[Binomial[4*j + 2, j] * Binomial[4*(n - j), n - j]/((4*j + 2)*(n - j + 1)), {j, 1, 2}] - 9 * Boole[n == 2]; Array[a, 22, 0] (* Amiram Eldar, May 12 2020 *)

[binomial(4*n + 1, n)/(4*n + 1) + 6*sum([binomial(4*j + 2, j)*binomial(4*(n - j), n - j)/(4*j + 2)/(n - j + 1) for j in srange(1, 3)]) - 9*(n==2) if n > 1 else 0 for n in srange(30)] # Benjamin Hackl, May 12 2020

a(0) = a(1) = 0 and a(n) = binomial(4*n+1, n)/(4*n+1) + 6*Sum_{j=1..2} binomial(4*j+2, j)*binomial(4*(n-j), n-j)/((4*j+2)*(n-j+1)) - 9*[n=2] for n > 1, where [ ] is the Iverson bracket.

A334649 a(n) is the total number of down steps between the third and fourth up steps in all 3_1-Dyck paths of length 4*n.

Original entry on oeis.org

0, 0, 0, 236, 1034, 6094, 40996, 295740, 2231022, 17370163, 138473536, 1124433142, 9266859394, 77307427741, 651540030688, 5538977450256, 47442103851930, 409000732566399, 3546232676711824, 30903652601552272, 270529448396053576, 2377829916885541565

Offset: 0

Benjamin Hackl, May 12 2020

A 3_1-Dyck path is a lattice path with steps (1, 3), (1, -1) that starts and ends at y = 0 and stays above the line y = -1.

For n = 3, there is no 4th up step, a(3) = 236 enumerates the total number of down steps between the 3rd up step and the end of the path.

Andrei Asinowski, Benjamin Hackl, Sarah J. Selkirk, Down-step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

Cf. A002293, A007226, A007228, A334645, A334646, A334647, A334648, A334680, A334682, A334785.

[binomial(4*n + 1, n)/(4*n + 1) + 6*sum([binomial(4*j + 2, j)*binomial(4*(n - j), n - j)/(4*j + 2)/(n - j + 1) for j in srange(1, 4)]) - 52*(n==3) if n > 2 else 0 for n in srange(30)] # Benjamin Hackl, May 12 2020

a(0) = a(1) = a(2) = 0 and a(n) = binomial(4*n+1, n)/(4*n+1) + 6*Sum_{j=1..3} binomial(4*j+2, j)*binomial(4*(n-j), n-j)/((4*j+2)*(n-j+1)) - 52*[n=3] for n > 2, where [ ] is the Iverson bracket.

A361033 a(n) = 3(4n)!/(n!*(n+1)!^3).

Original entry on oeis.org

3, 9, 280, 17325, 1513512, 162954792, 20193091776, 2768662192725, 409716429837000, 64358256798795960, 10605621798062141760, 1817833036248401270280, 321997225483126007438400, 58649494641569379926280000, 10941649720331183519046796800, 2084191938036600263793119045925

Offset: 0

Peter Bala, Mar 01 2023

Row 0 of A361032.
The central binomial numbers A000984(n) = (2*n)!/n!^2 have the property that A000984(n) is divisible by n + 1 and the result (2*n)!/(n!*(n+1)!) is the n-th Catalan number A000108(n). Similarly, the  numbers A008977(n) = (4*n)!/n!^4 appear to have the property that 3*A008977(n) is divisible by (n + 1)^3, leading to the present sequence. Cf. A361028. Do these numbers have a combinatorial interpretation?
Conjecture: a(n) is odd iff n = 2^k - 1 for some k >= 0.

Cf. A000108, A007226, A007228, A008977, A361028, A361032, A361034, A361035.

Maple
```
seq(3*(4*n)!/(n!*(n+1)!^3), n = 0..20);
```

Table[3 (4n)!/(n! ((n+1)!)^3),{n,0,15}] (* Harvey P. Dale, Jul 30 2024 *)

a(n) = 3*A008977(n)/(n+1)^3.
a(n) = (3/4)*A008977(n+1)/((4*n+1)*(4*n+2)*(4*n+3)).
a(n) = (1/2)*A007228(n)*A007226(n)*A000108(n).
P-recursive: a(n) = 4*(4*n-1)*(4*n-2)*(4*n-3)/(n+1)^3 * a(n-1) with a(0) = 3.
The o.g.f. A(x) satisfies the differential equation
x^3*(1 - 256*x)*A(x)''' + x^2*(6 - 1152*x)*A(x)'' + x*(7 - 816*x)*A(x)' + (1 - 24*x)*A(x) - 3 = 0 with A(0) = 3, A'(0) = 9 and A''(0) = 560.
a(n) ~ 3*sqrt(1/(2*Pi^3)) * 2^(8*n)/n^(9/2).

A361038 a(n) = 1680 * (3n)!/((2n)!*(n+3)!).

Original entry on oeis.org

280, 210, 420, 1176, 3960, 15015, 61880, 271320, 1248072, 5965050, 29414700, 148874400, 770263200, 4061212722, 21765976680, 118336861720, 651555929640, 3627981880950, 20405547069180, 115815267149400, 662742214356600

Offset: 0

Peter Bala, Mar 04 2023

Compare with the super ballot numbers A007272(n) = 60*(2*n)!/(n!*(n+3)!).

Cf. A000139, A001764, A007226, A007272, A361037, A361039, A361040.

seq( 1680 * (3*n)!/((2*n)!*(n+3)!), n = 0..20);

a(n) = 280*binomial(3*n,n) - 228*binomial(3*n,n+1) + 54*binomial(3*n,n+2) - 5*binomial(3*n,n+3). Thus a(n) is an integer.
P-recursive: 2*(n + 3)*(2*n - 1) = 3*(3*n - 1)*(3*n - 2)*a(n-1) with a(0) = 280.
a(n) ~ (27/4)^n * 840*sqrt(3/Pi)/n^(7/2).
The o.g.f. satisfies the differential equation
x^2*(27*x - 4)*A''(x) + 2*x*(27*x - 7)*A'(x) + (6*x + 6)*A(x) - 1680 = 0, with A(0) = 280 and A'(0) = 210.

A361039 a(n) = 55440 * (3n)!/((2n)!*(n+4)!).

Original entry on oeis.org

2310, 1386, 2310, 5544, 16335, 55055, 204204, 813960, 3432198, 15142050, 69334650, 327523680, 1588667850, 7883530578, 39904290580, 205532444040, 1075067283906, 5701114384350, 30608320603770, 166169731127400, 911270544740325

Offset: 0

Peter Bala, Mar 04 2023

Compare with the super ballot numbers A348893(n) = 840*(2*n)!/(n!*(n+4)!).

Cf. A000139, A001764, A007226, A007272, A361037, A361038, A361041.

seq(  55440 * (3*n)!/((2*n)!*(n+4)!), n = 0..20);

a(n) = 2310*binomial(3*n,n) - 2057*binomial(3*n,n+1) + 627*binomial(3*n,n+2) - 102*binomial(3*n,n+3) + 7*binomial(3*n, n+4). Thus a(n) is an integer.
P-recursive: 2*(n + 4)*(2*n - 1) = 3*(3*n - 1)*(3*n - 2)*a(n-1) with a(0) = 2310.
a(n) ~ (27/4)^n * 27720*sqrt(3/Pi)/n^(9/2).
The o.g.f. satisfies the differential equation
x^2*(27*x - 4)*A''(x) + 2*x*(27*x - 9)*A'(x) + (6*x + 8)*A(x) - 18480 = 0, with A(0) = 2310 and A'(0) = 1386.

A361037 a(n) = 20(3n)!/((2n)!(n+2)!).

Original entry on oeis.org

10, 10, 25, 84, 330, 1430, 6630, 32300, 163438, 852150, 4552275, 24812400, 137547000, 773564328, 4405019090, 25357898940, 147375745990, 863805209750, 5101386767295, 30332569967700, 181465130121450, 1091677288630950

Offset: 0

Peter Bala, Mar 04 2023

Gessel (1992) introduced sequences {b(r,n): n >= 0} of super ballot numbers defined by b(r,n) = J(r) * (2*n)!/(n!*(n + r + 1)!), r = 0,1,2,..., where J(r) = (2*r + 2)!/(2*(r + 1)!) = (2^r)*Product_{j = 0..r} (2*j + 1) is chosen so that these numbers are always integers. The sequence {b(1,n) : n >= 0} is A000108, the sequence of Catalan numbers. See A135573 for a table of these generalized Catalan numbers.

We carry out an analogous construction using the numbers B(n) = A005809(n) = binomial(3*n,n) = (3*n)!/((2*n)!*n!) in place of the central binomial numbers. We define B(r,n), r = 0,1,2, ..., by B(r,n) = F(r) * (3*n)!/((2*n)!*(n + r + 1)!), where F(r) is the minimal choice to produce integer values for these quantities for all n. This sequence is the case r = 1. See A007226 (r = 0), A361038 (r = 2) and A361039 (r = 3).

Ira M. Gessel, Super ballot numbers, J. Symbolic Comp., 14 (1992), 179-194.

Cf. A000139, A001764, A005809, A007226, A361038, A361039.

seq( 20*(3*n)!/((2*n)!*(n+2)!), n = 0..20);

Table[20 (3n)!/((2n)!(n+2)!),{n,0,30}] (* Harvey P. Dale, Aug 05 2024 *)

a(n) = 10*binomial(3*n,n) - 7*binomial(3*n,n+1) + binomial(3*n,n+2). Thus a(n) is an integer.
P-recursive: 2*(n + 2)(2*n - 1)*a(n) = 3*(3*n - 1)*(3*n - 2)*a(n-1) with a(0) = 10.
a(n) ~ (27/4)^n * 10*sqrt(3/Pi)/n^(5/2).
The o.g.f. satisfies the differential equation
x^2*(27*x - 4)*A''(x) + 2*x*(27*x - 5)*A'(x) + 2*(3*x + 2)*A(x) - 40 = 0, with A(0) = 10 and A'(0) = 10.

A334641 a(n) is the total number of down steps between the 3rd and 4th up steps in all 2-Dyck paths of length 3*n.

Original entry on oeis.org

0, 0, 0, 43, 108, 444, 2099, 10683, 56994, 314296, 1776519, 10236081, 59892690, 354886920, 2125117332, 12839859620, 78176677734, 479177993904, 2954360065247, 18309779343549, 114001476318240, 712751759478780, 4472908385838795, 28165267333869435

Offset: 0

Benjamin Hackl, May 07 2020

A 2-Dyck path is a nonnegative lattice path with steps (1, 2), (1, -1) that starts and ends at y = 0.

For n = 3, there is no 4th up step, a(3) = 43 enumerates the total number of down steps between the 3rd up step and the end of the path.

Andrei Asinowski, Benjamin Hackl, Sarah J. Selkirk, Down-step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

Cf. A007226, A007228, A124724, A334640, A334642, A334682.

a[0] = a[1] = a[2] = 0; a[n_] := 2 * Sum[Binomial[3*j + 1, j] * Binomial[3*(n - j), n - j]/((3*j + 1)*(n - j + 1)), {j, 1, 3}]; Array[a, 24, 0] (* Amiram Eldar, May 09 2020 *)

a(n) = if (n<=2, 0, 2*sum(j=1, 3, binomial(3*j+1, j)*binomial(3*(n-j), n-j)/((3*j+1)*(n-j+1)))); \\ Michel Marcus, May 09 2020

a(0) = a(1) = a(2) = 0 and a(n) = 2*Sum_{j=1..3}binomial(3*j+1, j)*binomial(3*(n-j), n-j)/((3*j+1)*(n-j+1)) for n > 2.

A334643 a(n) is the total number of down steps between the second and third up steps in all 2_1-Dyck paths of length 3*n. A 2_1-Dyck path is a lattice path with steps (1, 2), (1, -1) that starts and ends at y = 0 and stays above the line y = -1.

Original entry on oeis.org

0, 0, 16, 53, 209, 963, 4816, 25367, 138531, 777041, 4449511, 25901655, 152818458, 911755012, 5491420104, 33343242196, 203881825163, 1254342228285, 7759025239189, 48227078649155, 301056318504165, 1886647802277315, 11864793375611820, 74854437302309175

Offset: 0

Benjamin Hackl, May 12 2020

For n = 2, there is no 3rd up step, a(2) = 16 enumerates the total number of down steps between the 2nd up step and the end of the path.

			For n = 2, the 2_1-Dyck paths are UUDDDD, UDUDDD, UDDUDD, UDDDUD, DUDDUD, DUDUDD, DUUDDD. In total, there are a(2) = 4 + 3 + 2 + 1 + 1 + 2 + 3 = 16 down steps between the 2nd up step and the end of the path.

A. Asinowski, B. Hackl, and S. Selkirk, Down step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

Cf. A001764, A007226, A030983, A334640, A334642, A334644.

[binomial(3*n + 1, n)/(3*n + 1) + 4*sum([binomial(3*j + 2, j)*binomial(3*(n - j), n - j)/(3*j + 2)/(n - j + 1) for j in srange(1, 3)]) - 7*(n==2) if n >= 2 else 0 for n in srange(30)] # Benjamin Hackl, May 12 2020

a(0) = a(1) = 0 and a(n) = binomial(3*n+1, n)/(3*n+1) + 4*Sum_{j=1..2}binomial(3*j+2, j)*binomial(3*(n-j), n-j)/((3*j+2)*(n-j+1)) - 7*[n=2] for n > 1, where [ ] is the Iverson bracket.

A121446 Number of ternary trees with n edges and such that the first leaf in the preorder traversal is at level 1.

Original entry on oeis.org

3, 3, 10, 42, 198, 1001, 5304, 29070, 163438, 937365, 5462730, 32256120, 192565800, 1160346492, 7048030544, 43108428198, 265276342782, 1641229898525, 10202773534590, 63698396932170, 399223286267190, 2510857763851185, 15842014607109600, 100244747986099080

Offset: 1

Emeric Deutsch, Jul 30 2006

A ternary tree is a rooted tree in which each vertex has at most three children and each child of a vertex is designated as its left or middle or right child.

			a(1) = 3 because we have the trees /, | and \.
a(2) = 3 because we have the trees /|, /\ and |\.

Ira Gessel and Guoce Xin, The generating function of ternary trees and continued fractions, arXiv:math/0505217 [math.CO], 2005.
Ira Gessel and Guoce Xin, The generating function of ternary trees and continued fractions, Electronic Journal of Combinatorics, 13(1) (2006), #R53.

Cf. A007226, A001764, A006013, A006632.

Column 1 of A121445.

a:=proc(n) if n=1 then 3 else (2/n)*binomial(3*n-3,n-1) fi end: seq(a(n),n=1..25);

a[1] = 3; a[n_] := (2/n) Binomial[3 n - 3, n - 1];
Array[a, 22] (* Jean-François Alcover, Nov 28 2017 *)

a(n) = A007226(n-1) for n >= 2.
a(1) = 3 and a(n) = (2/n)*binomial(3*n-3, n-1) for n >= 2.
G.f.: (h - 1 - z)/(h - 1), where h = 1 + z*h^3 = 2*sin(arcsin(sqrt(27*z/4))/3)/sqrt(3*z).
D-finite with recurrence 2*n*(2*n - 3)*a(n) - 3*(3*n - 4)*(3*n - 5)*a(n-1) = 0 for n >= 3. - R. J. Mathar, Jun 22 2016
G.f.: 1-(1-(4*sin(arcsin((3^(3/2)*sqrt(x))/2)/3)^2)/3)^3. - Vladimir Kruchinin, Oct 04 2022
From Peter Bala, Jul 24 2025: (Start)
The g.f. A(x) = 3*x + 3*x^2 + 10*x^3 + ... satisfies the algebraic equation A(x)^3 - (3*x + 2)*A(x)^2 + (3*x^2 + 6*x + 1)*A(x) - (x^3 + 3*x^2 + 3*x) = 0.
1 + x/(1 - A(x)) = 1 + x + 3*x^2 + 12*x^3 + 55*x^4 + ... is the g.f. of A001764.
The g.f. A(x) satisfies (and is uniquely determined by) the conditions [x^n] (A(x) - 1)^n = 3 for n >= 1. (End)

A334608 a(n) is the total number of down-steps after the final up-step in all 3_1-Dyck paths of length 4*n (n up-steps and 3n down-steps).

Original entry on oeis.org

0, 5, 34, 236, 1714, 12922, 100300, 796572, 6443536, 52909593, 439896626, 3695917940, 31331587252, 267669458420, 2302188456120, 19918434257052, 173240112503520, 1513821095788420, 13283883136738344, 117009704490121520, 1034217260142108570, 9169842145476773250, 81537271617856588380

Offset: 0

Andrei Asinowski, May 13 2020

A 3_1-Dyck path is a lattice path with steps U=(1, 3), d=(1, -1) that starts at (0,0), stays (weakly) above y=-1, and ends at the x-axis.

			For n=1, a(1)=5 is the total number of down-steps after the last up-step in Uddd, dUdd.

A. Asinowski, B. Hackl, and S. Selkirk, Down step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

Cf. A002293, A007226, A007228, A334609, A334645, A334646, A334647, A334648, A334649, A334680, A334682, A334785.

a[n_] := 2 * Binomial[4*n + 6, n + 1]/(4*n + 6) - 4 * Binomial[4*n + 2, n]/(4*n + 2); Array[a, 23, 0] (* Amiram Eldar, May 13 2020 *)

[2*binomial(4*(n + 1) + 2, n + 1)/(4*(n + 1) + 2) - 4*binomial(4*n + 2, n)/(4*n + 2) for n in srange(30)] # Benjamin Hackl, May 13 2020

a(n) = 2*binomial(4*(n+1)+2, n+1)/(4*(n+1)+2) - 4*binomial(4*n+2, n)/(4*n+2).

cp's OEIS Frontend

A334648 a(n) is the total number of down steps between the second and third up steps in all 3_1-Dyck paths of length 4*n.

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A334649 a(n) is the total number of down steps between the third and fourth up steps in all 3_1-Dyck paths of length 4*n.

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A361033 a(n) = 3*(4*n)!/(n!*(n+1)!^3).

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A361038 a(n) = 1680 * (3*n)!/((2*n)!*(n+3)!).

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A361039 a(n) = 55440 * (3*n)!/((2*n)!*(n+4)!).

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A361037 a(n) = 20*(3*n)!/((2*n)!*(n+2)!).

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A334641 a(n) is the total number of down steps between the 3rd and 4th up steps in all 2-Dyck paths of length 3*n.

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A334643 a(n) is the total number of down steps between the second and third up steps in all 2_1-Dyck paths of length 3*n. A 2_1-Dyck path is a lattice path with steps (1, 2), (1, -1) that starts and ends at y = 0 and stays above the line y = -1.

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A361033 a(n) = 3(4n)!/(n!*(n+1)!^3).

A361038 a(n) = 1680 * (3n)!/((2n)!*(n+3)!).

A361039 a(n) = 55440 * (3n)!/((2n)!*(n+4)!).

A361037 a(n) = 20(3n)!/((2n)!(n+2)!).