A006631 -id:A006631 - cp's OEIS frontend

A006629 Self-convolution 4th power of A001764, which enumerates ternary trees.

1, 4, 18, 88, 455, 2448, 13566, 76912, 444015, 2601300, 15426840, 92431584, 558685348, 3402497504, 20858916870, 128618832864, 797168807855, 4963511449260, 31032552351570, 194743066471800, 1226232861415695

Offset: 0

Simon Plouffe, N. J. A. Sloane

Sum of root degrees of all noncrossing trees on nodes on a circle. - Emeric Deutsch

H. M. Finucan, Some decompositions of generalized Catalan numbers, pp. 275-293 of Combinatorial Mathematics IX. Proc. Ninth Australian Conference (Brisbane, August 1981). Ed. E. J. Billington, S. Oates-Williams and A. P. Street. Lecture Notes Math., 952. Springer-Verlag, 1982.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 0..200
Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.
Joris Nieuwveld, Fractions, Functions and Folding. A Novel Link between Continued Fractions, Mahler Functions and Paper Folding, Master's Thesis, arXiv:2108.11382 [math.NT], 2021.
C. H. Pah, Single polygon counting on Cayley Tree of order 3, J. Stat. Phys. 140 (2010) 198-207.
Index entries for sequences related to rooted trees

Column 2 of A092276.

Cf. A000139, A001764, A006013, A006630, A006631, A230547, A233657.

A006629:= func< n | 2*Binomial(3*n+3,n)/(n+2) >;
[A006629(n): n in [0..40]]; // G. C. Greubel, Aug 29 2025

Table[2*Binomial[3*n+3,n]/(n+2), {n,0,40}] (* G. C. Greubel, Aug 29 2025 *)

a(n)=my(m=4);binomial(3*n+m-1,n)*m/(2*n+m) /* 4th power of A001764 with offset n=0 */ \\ Paul D. Hanna, May 10 2008

def A006629(n): return 2*binomial(3*n+3,n)//(n+2)
print([A006629(n) for n in range(41)]) # G. C. Greubel, Aug 29 2025

a(n) = 2*binomial(3*n+3,n)/(n+2). - Emeric Deutsch
a(n) = (n+1) * A000139(n+1). - F. Chapoton, Feb 23 2024
G.f.: hypergeom( [ 2, 5/3, 4/3 ]; [ 3, 5/2 ]; 27*x/4 ).
G.f.: A(x) = G(x)^4 where G(x) = 1 + x*G(x)^3 = g.f. of A001764 giving a(n)=C(3n+m-1,n)*m/(2n+m) at power m=4 with offset n=0. - Paul D. Hanna, May 10 2008
G.f.: (((4*sin(arcsin((3*sqrt(3*x))/2)/3))/(sqrt(3*x))-1)^2-1)/(4*x). - Vladimir Kruchinin, Feb 17 2023
E.g.f.: hypergeom([4/3, 5/3, 2]; [1, 5/2, 3]; 27*x/4). - G. C. Greubel, Aug 29 2025

More precise definition from Paul D. Hanna, May 10 2008

A092276 Triangle read by rows: T(n,k) is the number of noncrossing trees with root degree equal to k.

Original entry on oeis.org

1, 2, 1, 7, 4, 1, 30, 18, 6, 1, 143, 88, 33, 8, 1, 728, 455, 182, 52, 10, 1, 3876, 2448, 1020, 320, 75, 12, 1, 21318, 13566, 5814, 1938, 510, 102, 14, 1, 120175, 76912, 33649, 11704, 3325, 760, 133, 16, 1, 690690, 444015, 197340, 70840, 21252, 5313, 1078, 168, 18, 1

Offset: 1

Emeric Deutsch, Feb 24 2004

With offset 0, Riordan array (f(x), x*f(x)) where f(x) is the g.f. of A006013. - Philippe Deléham, Jan 23 2010

			Triangle begins:
     1;
     2,    1;
     7,    4,    1;
    30,   18,    6,   1;
   143,   88,   33,   8,  1;
   728,  455,  182,  52, 10,  1;
  3876, 2448, 1020, 320, 75, 12, 1;
  ...
Top row of M^3 = (30, 18, 6, 1)
From _Peter Bala_, Nov 25 2024: (Start)
The transposed array as an infinite product of upper triangular arrays:
  /1 2 3 4 5 ... \/1            \/1              \       /1 2 7 30 143 ...\
  |  1 2 3 4 ... ||  1 2 3 4 ...||  1            |       |  1 4 18  88 ...|
  |    1 2 3 ... ||    1 2 3 ...||    1 2 3 4 ...| ... = |    1  6  33 ...|
  |      1 2 ... ||      1 2 ...||      1 2 3 ...|       |       1   8 ...|
  |        1 ... ||        1 ...||        1 2 ...|       |           1 ...|
  |          ... ||          ...||            ...|       |             ...|
Cf. A078812. (End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Peter Bala, Factorisations of some Riordan arrays as infinite products
Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.
Paul Barry, The second production matrix of a Riordan array, arXiv:2011.13985 [math.CO], 2020.
P. Flajolet and M. Noy, Analytic combinatorics of non-crossing configurations, Discrete Math., 204, 203-229, 1999.
M. Noy, Enumeration of noncrossing trees on a circle, Discrete Math., 180, 301-313, 1998.

Row sums give sequence A001764.

Columns 1..5 are A006013, A006629, A006630, A006631, A233657.

T := proc(n,k) if k=n then 1 else 2*k*binomial(3*n-k,n-k)/(3*n-k) fi end: seq(seq(T(n,k),k=1..n),n=1..11);

t[n_, n_] = 1; t[n_, k_] := 2*k*Binomial[3*n-k, n-k]/(3*n-k); Table[t[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 22 2012, after Maple *)

T(n, k) = 2*k*binomial(3*n-k, n-k)/(3*n-k); \\ Andrew Howroyd, Nov 06 2017

T(n, k) = 2*k*binomial(3n-k, n-k)/(3n-k).
G.f.: 1/(1-t*z*g^2), where g := 2*sin(arcsin(3*sqrt(3*z)/2)/3)/sqrt(3*z) is the g.f. of the sequence A001764.
T(n, k) = Sum_{j>=1} j*T(n-1, k-2+j). - Philippe Deléham, Sep 14 2005
With offset 0, T(n,k) = ((n+1)/(k+1))*binomial(3n-k+1, n-k). - Philippe Deléham, Jan 23 2010
From Gary W. Adamson, Jul 07 2011: (Start)
Let M = the production matrix
  2, 1;
  3, 2, 1;
  4, 3, 2, 1;
  5, 4, 3, 2, 1;
  ...
Top row of M^(n-1) generates n-th row terms of triangle A092276. Leftmost terms of each row = A006013 starting (1, 2, 7, 30, 143, ...). (End)
Working with an offset of 0, the inverse array is the Riordan array ((1 - x)^2, x*(1 - x)^2). - Peter Bala, Apr 30 2024

A006630 From generalized Catalan numbers.

Original entry on oeis.org

1, 6, 33, 182, 1020, 5814, 33649, 197340, 1170585, 7012200, 42364476, 257854776, 1579730984, 9734161206, 60290077905, 375138262520, 2343880406595, 14699630061270, 92502956574105, 583920410197950, 3696470074992240, 23461536762704040, 149270218961671548

Offset: 0

Simon Plouffe

It appears that this is the self-convolution of A001764 starting 1, 3, 12, ... . - Alon Regev, Aug 07 2015

H. M. Finucan, Some decompositions of generalized Catalan numbers, pp. 275-293 of Combinatorial Mathematics IX. Proc. Ninth Australian Conference (Brisbane, August 1981). Ed. E. J. Billington, S. Oates-Williams and A. P. Street. Lecture Notes Math., 952. Springer-Verlag, 1982.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Andrew Howroyd, Table of n, a(n) for n = 0..200
Alin Bostan, Frédéric Chyzak, and Vincent Pilaud, Refined product formulas for Tamari intervals, arXiv:2303.10986 [math.CO], 2023.
Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.

Column 3 of A092276.
Closely related to A000139.
Cf. A001764, A006631.

[2*Binomial(3*n+6, n)/(n+2): n in [0..25]]; // Vincenzo Librandi, Aug 07 2015

Table[2 Binomial[3 n+6,n]/(n+2), {n, 0, 25}] (* Vincenzo Librandi, Aug 07 2015 *)
CoefficientList[Series[(-1 + (2*Sin[(1/3)*ArcSin[(3*Sqrt[3]*Sqrt[x])/2]]) / (Sqrt[3]*Sqrt[x]))^2/x^2, {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 03 2022, after Vladimir Kruchinin *)

taylor(((1/sqrt(3/4*x)*sin(1/3*asin(sqrt(27/4*x)))-1)/x)^2,x,0,17); /* Vladimir Kruchinin, Oct 03 2022 */

makelist(2*binomial(3*n+6, n)/(n+2),n,0,30); /* Vladimir Kruchinin, Oct 03 2022 */

a(n) = 2*binomial(3*n+6, n)/(n+2); \\ Andrew Howroyd, Nov 06 2017

def A006630(n): return 2*binomial(3*(n+2),n)//(n+2)
print([A006630(n) for n in range(41)]) # G. C. Greubel, Aug 31 2025

G.f.: hypergeometric3_F_2([ 2, 8/3, 7/3 ], [ 4, 7/2 ], 27*x/4).
a(n) = 2*binomial(3*n+6, n)/(n+2). - Henry Bottomley, Sep 24 2001
G.f.: (1 - RootOf(x-t*(1-t)^2,t))^(-6) (algebraic function in Maple notation). - Mark van Hoeij, Nov 08 2011
G.f.: ((1/sqrt((3/4)*x)*sin((1/3)*asin(sqrt((27/4)*x)))-1)/x)^2. - Vladimir Kruchinin, Oct 03 2022
a(n) = (n+1)/2 * A000139(n+2). - F. Chapoton, Feb 23 2024

More terms from Christopher Lund (clund(AT)san.rr.com), Apr 16 2002

a(21)-a(22) from Vincenzo Librandi, Aug 07 2015

A006634 a(n) = 3binomial(4n+8, n)/(n+3).

Original entry on oeis.org

1, 9, 72, 570, 4554, 36855, 302064, 2504304, 20974005, 177232627, 1509395976, 12943656180, 111676661460, 968786892675, 8445123522144, 73940567860896, 649942898236596, 5733561315124260, 50744886833898400, 450461491952952690, 4009721145437152530, 35782256673785401065

Offset: 0

Simon Plouffe

Former name: From generalized Catalan numbers.

H. M. Finucan, Some decompositions of generalized Catalan numbers, pp. 275-293 of Combinatorial Mathematics IX. Proc. Ninth Australian Conference (Brisbane, August 1981). Ed. E. J. Billington, S. Oates-Williams and A. P. Street. Lecture Notes Math., 952. Springer-Verlag, 1982.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Simon Plouffe, Approximations of generating functions and a few conjectures, arXiv:0911.4975 [math.NT], 2002.

Cf. A006630, A006631, A006632, A006633, A006635, A006636, A006637.

A006634:= func< n | 3*Binomial(4*n+8,n)/(n+3) >;
[A006634(n): n in [0..40]]; // G. C. Greubel, Sep 01 2025

series(RootOf(g = 1+x*g^4,g)^9, x=0, 30); # Mark van Hoeij, Apr 22 2013

f[x_] := HypergeometricPFQ[ {9/4, 5/2, 11/4, 3}, {10/3, 11/3, 4}, 256/27*x]; Series[f[x], {x, 0, 16}] // CoefficientList[#, x]& (* Jean-François Alcover, Apr 23 2013, after Simon Plouffe *)
Table[3*Binomial[4*n+8,n]/(n+3), {n,0,40}] (* G. C. Greubel, Sep 01 2025 *)

N = 3*66;  x = 'x + O('x^N);
g=serreverse(x-x^4)/x;
gf=g^9;  v=Vec(gf);
vector(#v\3,n,v[3*n-2])
/* Joerg Arndt, Apr 23 2013 */

def A006634(n): return 3*binomial(4*(n+2),n)//(n+3)
print([A006634(n) for n in range(41)]) # G. C. Greubel, Sep 01 2025

G.f.: hypergeom([9/4, 5/2, 11/4, 3], [10/3, 11/3, 4], 256/27*x). - Simon Plouffe, Master's Thesis, UQAM, 1992
G.f.: g^9 where g = 1+x*g^4 is the g.f. of A002293. - Mark van Hoeij, Apr 22 2013
From G. C. Greubel, Sep 01 2025: (Start)
a(n) = 3*binomial(4*n+8, n)/(n+3).
E.g.f.: hypergeom([9/4, 5/2, 11/4, 3], [1, 10/3, 11/3, 4], 256*x/27). (End)

More terms from Joerg Arndt, Apr 23 2013

New name by G. C. Greubel, Sep 01 2025

A006633 Expansion of hypergeom([3/2, 7/4, 2, 9/4], [7/3, 8/3, 3], (256/27)*x).

Original entry on oeis.org

1, 6, 39, 272, 1995, 15180, 118755, 949344, 7721604, 63698830, 531697881, 4482448656, 38111876530, 326439471960, 2814095259675, 24397023508416, 212579132600076, 1860620845932216, 16351267454243260, 144222309948974400, 1276307560533365955, 11329053395044653180

Offset: 0

Simon Plouffe

nonn

From generalized Catalan numbers.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Paolo Xausa, Table of n, a(n) for n = 0..1000
H. M. Finucan, Some decompositions of generalized Catalan numbers, pp. 275-293 of Combinatorial Mathematics IX. Proc. Ninth Australian Conference (Brisbane, August 1981). Ed. E. J. Billington, S. Oates-Williams and A. P. Street. Lecture Notes Math., 952. Springer-Verlag, 1982.
Simon Plouffe, Approximations of generating functions and a few conjectures, Master's Thesis, arXiv:0911.4975 [math.NT], 2009.

Cf. A006631, A006632, A006634, A006635, A000027, A000260.

gf := hypergeom([3/2, 7/4, 2, 9/4], [7/3, 8/3, 3], (256/27)*x):
ser := series(gf, x, 22): seq(coeff(ser, x, n), n = 0..21); # Peter Luschny, Feb 22 2024

A006633[n_] := 2*Binomial[4*n+5, n]/(n+2);
Array[A006633, 25, 0] (* Paolo Xausa, Feb 25 2024 *)

O.g.f.: hypergeom_4F3([3/2, 7/4, 2, 9/4], [7/3, 8/3, 3], (256/27)*x). - Simon Plouffe, Master's Thesis, UQAM 1992
a(n) = 2*binomial(4*n + 5, n) / (n+2). - Bruno Berselli, Jan 18 2014
a(n) = (n+1) * A000260(n+1). - F. Chapoton, Feb 22 2024

New name by using a formula from the author by Peter Luschny, Feb 24 2024

A110616 A convolution triangle of numbers based on A001764.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 12, 7, 3, 1, 55, 30, 12, 4, 1, 273, 143, 55, 18, 5, 1, 1428, 728, 273, 88, 25, 6, 1, 7752, 3876, 1428, 455, 130, 33, 7, 1, 43263, 21318, 7752, 2448, 700, 182, 42, 8, 1, 246675, 120175, 43263, 13566, 3876, 1020, 245, 52, 9, 1

Offset: 0

Philippe Deléham, Sep 14 2005, Jun 15 2007

Reflected version of A069269. - Vladeta Jovovic, Sep 27 2006
With offset 1 for n and k, T(n,k) = number of Dyck paths of semilength n for which all descents are of even length (counted by A001764) with no valley vertices at height 1 and with k returns to ground level. For example, T(3,2)=2 counts U^4 D^4 U^2 D^2, U^2 D^2 U^4 D^4 where U=upstep, D=downstep and exponents denote repetition. - David Callan, Aug 27 2009
Riordan array (f(x), x*f(x)) with f(x) = (2/sqrt(3*x))*sin((1/3)*arcsin(sqrt(27*x/4))). - Philippe Deléham, Jan 27 2014
Antidiagonals of convolution matrix of Table 1.4, p. 397, of Hoggatt and Bicknell. - Tom Copeland, Dec 25 2019

			Triangle begins:
       1;
       1,      1;
       3,      2,     1;
      12,      7,     3,     1;
      55,     30,    12,     4,    1;
     273,    143,    55,    18,    5,    1;
    1428,    728,   273,    88,   25,    6,   1;
    7752,   3876,  1428,   455,  130,   33,   7,  1;
   43263,  21318,  7752,  2448,  700,  182,  42,  8, 1;
  246675, 120175, 43263, 13566, 3876, 1020, 245, 52, 9, 1;
  ...
From _Peter Bala_, Feb 04 2025: (Start)
The transposed array factorizes as an infinite product of upper triangular arrays:
  / 1               \^T   /1             \^T /1             \^T / 1            \^T
  | 1    1           |   | 1   1          | | 0  1           |  | 0  1          |
  | 3    2   1       | = | 2   1   1      | | 0  1   1       |  | 0  0  1       | ...
  |12    7   3   1   |   | 5   2   1  1   | | 0  2   1  1    |  | 0  0  1  1    |
  |55   30  12   4  1|   |14   5   2  1  1| | 0  5   2  1  1 |  | 0  0  2  1  1 |
  |...               |   |...             | |...             |  |...            |
where T denotes transposition and [1, 1, 2, 5, 14,...] is the sequence of Catalan numbers A000108. (End)

Peter Bala, Factorisations of some Riordan arrays as infinite products
Paul Barry, Notes on Riordan arrays and lattice paths, arXiv:2504.09719 [math.CO], 2025. See pp. 27, 29.
Paul Barry, d-orthogonal polynomials, Fuss-Catalan matrices and lattice paths, arXiv:2505.16718 [math.CO], 2025. See p. 21.
Naiomi Cameron and J. E. McLeod, Returns and Hills on Generalized Dyck Paths, Journal of Integer Sequences, Vol. 19, 2016, #16.6.1.
V. E. Hoggatt, Jr. and M. Bicknell, Catalan and related sequences arising from inverses of Pascal's triangle matrices, Fib. Quart., 14 (1976), 395-405.
Sheng-Liang Yang and L. J. Wang, Taylor expansions for the m-Catalan numbers, Australasian Journal of Combinatorics, Volume 64(3) (2016), Pages 420-431.

Successive columns: A001764, A006013, A001764, A006629, A102893, A006630, A102594, A006631; row sums: A098746; see also A092276.

Table[(k + 1) Binomial[3 n - 2 k, 2 n - k]/(2 n - k + 1), {n, 0, 9}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jun 28 2017 *)

T(n,k):=((k+1)*binomial(3*n-2*k,2*n-k))/(2*n-k+1); /* Vladimir Kruchinin, Nov 01 2011 */

T(n, k) = Sum_{j>=0} T(n-1, k-1+j)*A000108(j); T(0, 0) = 1; T(n, k) = 0 if k < 0 or if k > n.
G.f.: 1/(1 - x*y*TernaryGF) = 1 + (y)x + (y+y^2)x^2 + (3y+2y^2+y^3)x^3 +... where TernaryGF = 1 + x + 3x^2 + 12x^3 + ... is the GF for A001764. - David Callan, Aug 27 2009
T(n, k) = ((k+1)*binomial(3*n-2*k,2*n-k))/(2*n-k+1). - Vladimir Kruchinin, Nov 01 2011

A230547 a(n) = 3binomial(3n+9, n)/(n+3).

Original entry on oeis.org

1, 9, 63, 408, 2565, 15939, 98670, 610740, 3786588, 23535820, 146710476, 917263152, 5752004349, 36174046743, 228124619100, 1442387942520, 9142452842985, 58083251802345, 369816259792035, 2359448984037600

Offset: 0

Tim Fulford, Oct 23 2013

nonn

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=3, r=9.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906 [math.CO], 2007; Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers Chapter 7
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.
Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.

Cf. A000108, A001764, A006013, A006629, A102893, A006630, A102594, A006631, A233657.

[9*Binomial(3*n+9, n)/(3*n+9): n in [0..30]];

Table[9 Binomial[3 n + 9, n]/(3 n + 9), {n, 0, 30}]

PARI
```
a(n) = 9*binomial(3*n+9,n)/(3*n+9);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(3/9))^9+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, here p=3, r=9.

D-finite with recurrence 2*n*(2*n+9)*(n+4)*a(n) -3*(3*n+7)*(n+2)*(3*n+8)*a(n-1)=0. - R. J. Mathar, Nov 22 2024

A233657 a(n) = 10 * binomial(3n+10,n)/(3n+10).

Original entry on oeis.org

1, 10, 75, 510, 3325, 21252, 134550, 848250, 5340060, 33622600, 211915132, 1337675430, 8458829925, 53591180360, 340185835500, 2163581913780, 13786238414025, 88004926973250, 562763873596575, 3604713725613000, 23126371951808268, 148594788106641360

Offset: 0

Tim Fulford, Dec 14 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=3, r=10.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
David Bevan, Robert Brignall, Andrew Elvey Price and Jay Pantone, A structural characterisation of Av(1324) and new bounds on its growth rate, arXiv preprint arXiv:1711.10325 [math.CO], 2017-2019.
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906 [math.CO], 2007; Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers Chapter 7
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955 (2010).
Emanuele Munarini, Shifting Property for Riordan, Sheffer and Connection Constants Matrices, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.2.

Column 5 of A092276.

Cf. A000108, A001764, A006013, A006629, A102893, A006630, A102594, A006631, A230547.

[10*Binomial(3*n+10, n)/(3*n+10): n in [0..30]];

A233657:=n->10*binomial(3*n+10,n)/(3*n+10): seq(A233657(n), n=0..20); # Wesley Ivan Hurt, Oct 10 2014

Table[10 Binomial[3 n + 10, n]/(3 n + 10), {n, 0, 30}]

PARI
```
a(n) = 10*binomial(3*n+10,n)/(3*n+10);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(3/10))^10+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, here p=3, r=10.
+2*n*(n+5)*(2*n+9)*a(n) -3*(3*n+7)*(n+3)*(3*n+8)*a(n-1)=0. - R. J. Mathar, Feb 16 2018
E.g.f.: F([10/3, 11/3, 4], [1, 11/2, 6], 27*x/4), where F is the generalized hypergeometric function. - Stefano Spezia, Oct 08 2019

A069269 Second level generalization of Catalan triangle (0th level is Pascal's triangle A007318; first level is Catalan triangle A009766; 3rd level is A069270).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 7, 12, 1, 4, 12, 30, 55, 1, 5, 18, 55, 143, 273, 1, 6, 25, 88, 273, 728, 1428, 1, 7, 33, 130, 455, 1428, 3876, 7752, 1, 8, 42, 182, 700, 2448, 7752, 21318, 43263, 1, 9, 52, 245, 1020, 3876, 13566, 43263, 120175, 246675

Offset: 0

Henry Bottomley, Mar 12 2002

For the m-th level generalization of Catalan triangle T(n,k) = C(n+mk,k)*(n-k+1)/(n+(m-1)k+1); for n >= k+m: T(n,k) = T(n-m+1,k+1) - T(n-m,k+1); and T(n,n) = T(n+m-1,n-1) = C((m+1)n,n)/(mn+1).
Reflected version of A110616. - Philippe Deléham, Jun 15 2007
With offset 1 for n and k, T(n,k) is (conjecturally) the number of permutations of [n] that avoid the patterns 4-2-3-1 and 4-2-5-1-3 and for which the last ascent ends at position k (k=1 if there are no ascents). For example, T(4,1) = 1 counts 4321; T(4,2) = 3 counts 1432, 2431, 3421; T(4,3) = 7 counts 1243, 1342, 2143, 2341, 3142, 3241, 4132. - David Callan, Jul 22 2008
Row sums appear to be in A098746. - R. J. Mathar, May 30 2014

			Rows start
  1;
  1,  1;
  1,  2,  3;
  1,  3,  7, 12;
  1,  4, 12, 30, 55;

M. H. Albert et al., Restricted permutations and queue jumping, Discrete Math., 287 (2004), 129-133.
Tad White, Quota Trees, arXiv:2401.01462 [math.CO], 2024. See p. 18.

Columns include A000012, A000027, A055998.
Right-hand diagonals include A001764, A006013, A006629, A006630, A006631.
Cf. triangles A007318, A009766, A069270.

T(n, k) = C(n+2k, k)*(n-k+1)/(n+k+1).
For n >= k+2: T(n, k) = T(n-1, k+1) - T(n-2, k+1).
T(n, n) = T(n+1, n-1) = C(3n, n)/(2n+1).

cp's OEIS Frontend

A006629 Self-convolution 4th power of A001764, which enumerates ternary trees.

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A092276 Triangle read by rows: T(n,k) is the number of noncrossing trees with root degree equal to k.

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A006630 From generalized Catalan numbers.

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A006634 a(n) = 3*binomial(4*n+8, n)/(n+3).

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A006633 Expansion of hypergeom([3/2, 7/4, 2, 9/4], [7/3, 8/3, 3], (256/27)*x).

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A110616 A convolution triangle of numbers based on A001764.

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A006634 a(n) = 3binomial(4n+8, n)/(n+3).

A230547 a(n) = 3binomial(3n+9, n)/(n+3).

A233657 a(n) = 10 * binomial(3n+10,n)/(3n+10).