A102594 -id:A102594 - cp's OEIS frontend

A110616 A convolution triangle of numbers based on A001764.

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

A143603 Triangle, read by rows, such that the g.f. of column k = G(x)^(2k+1) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764 (ternary trees).

Original entry on oeis.org

1, 1, 1, 3, 3, 1, 12, 12, 5, 1, 55, 55, 25, 7, 1, 273, 273, 130, 42, 9, 1, 1428, 1428, 700, 245, 63, 11, 1, 7752, 7752, 3876, 1428, 408, 88, 13, 1, 43263, 43263, 21945, 8379, 2565, 627, 117, 15, 1, 246675, 246675, 126500, 49588, 15939, 4235, 910, 150, 17, 1

Offset: 1

Paul D. Hanna, Aug 29 2008

From Peter Bala, Aug 07 2014: (Start)
Riordan array (G(x), x*G(x)). Let C(x) = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + ... be the o.g.f. of the Catalan numbers A000108. Then C(x*G(x)) = G(x).
This leads to a factorization of this array in the group of Riordan matrices as (1, x*G(x))*(C(x), x*C(x)) = (1 + A110616)*A033184 (here, in the final product, 1 refers to the 1 X 1 identity matrix and + means direct sum - see the Example section). (End)

			Triangle begins:
1;
1, 1;
3, 3, 1;
12, 12, 5, 1;
55, 55, 25, 7, 1;
273, 273, 130, 42, 9, 1;
1428, 1428, 700, 245, 63, 11, 1;
7752, 7752, 3876, 1428, 408, 88, 13, 1; ...
where g.f. of column k = G(x)^(2k+1) where G(x) = 1 + x*G(x)^3.
Matrix inverse begins:
1;
-1, 1;
0, -3, 1;
0, 3, -5, 1;
0, -1, 10, -7, 1;
0, 0, -10, 21, -9, 1;
0, 0, 5, -35, 36, -11, 1;
0, 0, -1, 35, -84, 55, -13, 1; ...
where g.f. of column k = (1-x)^(2k+1) for k>=0.
From _Peter Bala_, Aug 07 2014: (Start)
Matrix factorization as (1 + A110616)*A033184 begins
/1           \/ 1         \    / 1           \
|0  1        || 1  1       |   | 1  1        |
|0  1 1      || 2  2 1     | = | 3  3  1     |
|0  3 2 1    || 5  5 3 1   |   |12 12  5 1   |
|0 12 7 3 1  ||14 14 9 4 1 |   |55 55 25 7 1 |
(End)

Yuxuan Zhang and Yan Zhuang, A subfamily of skew Dyck paths related to k-ary trees, arXiv:2306.15778 [math.CO], 2023.

Cf. columns: A001764, A102893, A102594; row sums: A006013. A033184, A110616.

{T(n,k)=binomial(3*n-k,n-k)*(2*k+1)/(2*n+1)}

T(n,k) = C(3n-k,n-k)*(2k+1)/(2n+1) for 0<=k<=n.
Let M = the production matrix:
1, 1
2, 2, 1
3, 3, 2, 1
4, 4, 3, 2, 1
5, 5, 4, 3, 2, 1
...
Top row of M^(n-1) gives n-th row. - Gary W. Adamson, Jul 07 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

A355172 The Fuss-Catalan triangle of order 2, read by rows. Related to ternary trees.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 1, 5, 12, 0, 1, 7, 25, 55, 0, 1, 9, 42, 130, 273, 0, 1, 11, 63, 245, 700, 1428, 0, 1, 13, 88, 408, 1428, 3876, 7752, 0, 1, 15, 117, 627, 2565, 8379, 21945, 43263, 0, 1, 17, 150, 910, 4235, 15939, 49588, 126500, 246675

Offset: 0

Peter Luschny, Jun 25 2022

The Fuss-Catalan triangle of order m is a regular, (0, 0)-based table recursively defined as follows: Set row(0) = [1] and row(1) = [0, 1]. Now assume row(n-1) already constructed and duplicate the last element of row(n-1). Next apply the cumulative sum m times to this list to get row(n). Here m = 2. (See the Python program for a reference implementation.)

This definition also includes the Fuss-Catalan numbers A001764(n) = T(n, n), or row 3 in A355262. For m = 1 see A355173 and for m = 3 A355174. More generally, for n >= 1 all Fuss-Catalan sequences (A355262(n, k), k >= 0) are the main diagonals of the Fuss-Catalan triangles of order n - 1.

			Table T(n, k) begins:
  [0] [1]
  [1] [0, 1]
  [2] [0, 1,  3]
  [3] [0, 1,  5, 12]
  [4] [0, 1,  7, 25,  55]
  [5] [0, 1,  9, 42, 130,  273]
  [6] [0, 1, 11, 63, 245,  700, 1428]
  [7] [0, 1, 13, 88, 408, 1428, 3876, 7752]
Seen as an array reading the diagonals starting from the main diagonal:
  [0] 1, 1,  3, 12,  55,  273,  1428,   7752,   43263,  246675, ...  A001764
  [1] 0, 1,  5, 25, 130,  700,  3876,  21945,  126500,  740025, ...  A102893
  [2] 0, 1,  7, 42, 245, 1428,  8379,  49588,  296010, 1781325, ...  A102594
  [3] 0, 1,  9, 63, 408, 2565, 15939,  98670,  610740, 3786588, ...  A230547
  [4] 0, 1, 11, 88, 627, 4235, 27830, 180180, 1157013, 7396972, ...

A001764 (main diagonal), A102893 (subdiagonal), A102594 (diagonal 2), A230547 (diagonal 3), A005408 (column 2), A071355 (column 3), A006629 (row sums), A143603 (variant).

Cf. A123110 (triangle of order 0), A355173 (triangle of order 1), A355174 (triangle of order 3), A355262 (Fuss-Catalan array).

from functools import cache
from itertools import accumulate
@cache
def Trow(n: int) -> list[int]:
    if n == 0: return [1]
    if n == 1: return [0, 1]
    row = Trow(n - 1) + [Trow(n - 1)[n - 1]]
    return list(accumulate(accumulate(row)))
for n in range(9): print(Trow(n))

The general formula for the Fuss-Catalan triangles is, for m >= 0 and 0 <= k <= n:
FCT(n, k, m) = (m*(n - k) + m + 1)*(m*n + k - 1)!/((m*n + 1)!*(k - 1)!) for k > 0 and FCT(n, 0, m) = 0^n. The case considered here is T(n, k) = FCT(n, k, 2).
T(n, k) = (2*n - 2*k + 3)*(2*n + k - 1)!/((2*n + 1)!*(k - 1)!) for k > 0; T(n, 0) = 0^n.
The g.f. of row n of the FC-triangle of order m is 0^n + (x - (m + 1)*x^2) / (1 - x)^(m*n + 2), thus:
T(n, k) = [x^k] (0^n + (x - 3*x^2)/(1 - x)^(2*n + 2)).

A102595 Triangle read by rows: T(n,k) is the number of noncrossing trees with n edges in which the maximal number of contiguous border edges starting from the root in both directions is equal to k.

Original entry on oeis.org

1, 0, 1, 0, 0, 3, 1, 4, 3, 4, 7, 20, 15, 8, 5, 42, 102, 72, 36, 15, 6, 245, 540, 366, 176, 70, 24, 7, 1428, 2950, 1944, 912, 355, 120, 35, 8, 8379, 16524, 10668, 4920, 1890, 636, 189, 48, 9, 49588, 94430, 60021, 27336, 10405, 3492, 1050, 280, 63, 10, 296010

Offset: 0

Emeric Deutsch, Jan 22 2005

Row sums yield the ternary numbers (A001764).

T(n,0) = A102594(n).

			T(2,0)=T(2,1)=0, T(2,2)=3 because in all the noncrossing trees _\, /\ and /_, the maximal number of contiguous border edges starting from the root in both directions is equal to 2.
Triangle starts:
   1;
   0,   1;
   0,   0,  3;
   1,   4,  3,  4;
   7,  20, 15,  8,  5;
  42, 102, 72, 36, 15, 6;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1274
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.

Cf. A001764, A102594.

G:=(g+z*g-t*z-2*z*g^2+t^2*(1-t)*z^3*g^2-2*t*(1-t)*z^2*g)/(1-t*z*g)^2: z:=w^2: b:=w*sqrt(3): g:=2*sin(arcsin(3*b/2)/3)/b: Gser:=simplify(series(G,w=0,24)): P[0]:=1: for n from 1 to 10 do P[n]:=sort(coeff(Gser,w^(2*n))) od: for n from 0 to 10 do seq(coeff(t*P[n],t^k),k=1..n+1) od; # yields sequence in triangular form

max = 20; z = w^2; b = w*Sqrt[3]; g = 2*(Sin[ ArcSin[3*(b/2)]/3]/b); gf = (g + z*g - t*z - 2*z*g^2 + t^2*(1 - t)*z^3*g^2 - 2*t*(1 - t)*z^2*g)/(1 - t*z*g)^2; se = Series[gf, {w, 0, max}]; Flatten[ Rest /@ DeleteCases[ (CoefficientList[t*#1, t] & ) /@ CoefficientList[se, w], {}]] (* Jean-François Alcover, Oct 05 2011, after Maple *)

S(n)={my(g=1+serreverse(x/(1+x)^3 + O(x*x^n))); Vec((g + x*g - y*x - 2*x*g^2 + y^2*(1-y)*x^3*g^2 - 2*y*(1-y)*x^2*g)/(1 - y*x*g)^2)}
my(v=S(10)); for(n=1, #v, my(p=v[n]); for(k=0, n-1, print1(polcoeff(p, k), ", ")); print); \\ Andrew Howroyd, Nov 17 2017

G.f.: G(t, z)=(g+zg-tz-2zg^2+t^2*(1-t)z^3*g^2-2t(1-t)z^2*g)/(1-tzg)^2, where g=1+zg^3 is the g.f. for the ternary numbers (A001764).

cp's OEIS Frontend

A110616 A convolution triangle of numbers based on A001764.

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A143603 Triangle, read by rows, such that the g.f. of column k = G(x)^(2k+1) where G(x) = 1 + x*G(x)^3 is the g.f. of A001764 (ternary trees).

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

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

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A355172 The Fuss-Catalan triangle of order 2, read by rows. Related to ternary trees.

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A102595 Triangle read by rows: T(n,k) is the number of noncrossing trees with n edges in which the maximal number of contiguous border edges starting from the root in both directions is equal to k.

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Maple

Mathematica

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

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