A007228 -id:A007228 - cp's OEIS frontend

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

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).

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.

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).

A334650 a(n) is the total number of down steps between the first and second up steps in all 3_2-Dyck paths of length 4*n.

Original entry on oeis.org

0, 6, 31, 158, 975, 6639, 48050, 362592, 2820789, 22460120, 182141553, 1499143282, 12490923757, 105150960654, 892973346300, 7640934031920, 65813450140017, 570160918044288, 4964875184429660, 43431741548248440, 381496856026500220, 3363457643008999635

Offset: 0

Benjamin Hackl, May 13 2020

A 3_2-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 = -2.

For n = 1, there is no 2nd up step, a(1) = 6 enumerates the total number of down steps between the 1st up step and the end of the path.

			For n = 1, the 3_2-Dyck paths are DDUD, DUDD, UDDD. This corresponds to a(1) = 1 + 2 + 3 = 6 down steps between the 1st 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, A007228, A334609, A334647, A334648, A334649, A334785.

a[0] = 0; a[n_] := 3 * Binomial[4*n, n]/(n + 1) - Binomial[4*n + 2, n]/(n + 1) + 9 * Binomial[4*(n - 1), n - 1]/n - 6 * Boole[n == 1]; Array[a, 22, 0] (* Amiram Eldar, May 13 2020 *)

[3*binomial(4*n, n)/(n + 1) - binomial(4*n + 2, n)/(n + 1) + 9*binomial(4*(n - 1), n - 1)/n - 6*(n==1) if n > 0 else 0 for n in srange(30)] # Benjamin Hackl, May 13 2020

a(0) = 0 and a(n) = 3*binomial(4*n, n)/(n+1) - binomial(4*n+2, n)/(n+1) + 9*binomial(4*(n-1), n-1)/n - 6*[n=1] for n > 0, where [ ] is the Iverson bracket.

A241262 Array t(n,k) = binomial(n*k, n+1)/n, where n >= 1 and k >= 2, read by ascending antidiagonals.

Original entry on oeis.org

1, 2, 3, 5, 10, 6, 14, 42, 28, 10, 42, 198, 165, 60, 15, 132, 1001, 1092, 455, 110, 21, 429, 5304, 7752, 3876, 1020, 182, 28, 1430, 29070, 57684, 35420, 10626, 1995, 280, 36, 4862, 163438, 444015, 339300, 118755, 24570, 3542, 408, 45, 16796, 937365, 3506100, 3362260, 1391280, 324632, 50344, 5850, 570, 55

Offset: 1

Jean-François Alcover, Apr 18 2014

About the "root estimation" question asked in MathOverflow, one can check (at least numerically) that, for instance with k = 4 and a = 1/11, the series a^-1 + (k - 1) + Sum_{n>=} (-1)^n*binomial(n*k, n+1)/n*a^n evaluates to the positive solution of x^k = (x+1)^(k-1).
Row 1 is A000217 (triangular numbers),
Row 2 is A006331 (twice the square pyramidal numbers),
Row 3 is A067047(3n) = lcm(3n, 3n+1, 3n+2, 3n+3)/12 (from column r=4 of A067049),
Row 4 is A222715(2n) = (n-1)*n*(2n-1)*(4n-3)*(4n-1)/15,
Row 5 is not in the OEIS.
Column 1 is A000108 (Catalan numbers),
Column 2 is A007226 left shifted 1 place,
Column 4 is A007228 left shifted 1 place,
Column 5 is A124724 left shifted 1 place,
Column 6 is not in the OEIS.

			Array begins:
    1,    3,     6,     10,      15,      21, ...
    2,   10,    28,     60,     110,     182, ...
    5,   42,   165,    455,    1020,    1995, ...
   14,  198,  1092,   3876,   10626,   24570, ...
   42, 1001,  7752,  35420,  118755,  324632, ...
  132, 5304, 57684, 339300, 1391280, 4496388, ...
  etc.

N. S. S. Gu, H. Prodinger, S. Wagner, Bijections for a class of labeled plane trees, Eur. J. Combinat. 31 (2010) 720-732, doi|10.1016/j.ejc.2009.10.007, Theorem 2

MathOverflow, Root estimation

Cf. A000217, A006331, A000108, A007226, A007228, A067047, A067049, A124724, A222715.

t[n_, k_] := Binomial[n*k, n+1]/n; Table[t[n-k+2, k], {n, 1, 10}, {k, 2, n+1}] // Flatten

cp's OEIS Frontend

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

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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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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).

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A334650 a(n) is the total number of down steps between the first and second up steps in all 3_2-Dyck paths of length 4*n.

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A241262 Array t(n,k) = binomial(n*k, n+1)/n, where n >= 1 and k >= 2, read by ascending antidiagonals.

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