A006629 -id:A006629 - 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

A167763 Pendular triangle (p=0), read by rows, where row n is formed from row n-1 by the recurrence: if n > 2k, T(n,k) = T(n,n-k) + T(n-1,k), otherwise T(n,k) = T(n,n-1-k) + p*T(n-1,k), for n >= k <= 0, with T(n,0) = 1 and T(n,n) = 0^n.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 1, 4, 7, 4, 1, 0, 1, 5, 12, 12, 5, 1, 0, 1, 6, 18, 30, 18, 6, 1, 0, 1, 7, 25, 55, 55, 25, 7, 1, 0, 1, 8, 33, 88, 143, 88, 33, 8, 1, 0, 1, 9, 42, 130, 273, 273, 130, 42, 9, 1, 0, 1, 10, 52, 182, 455, 728, 455, 182, 52, 10, 1, 0

Offset: 0

Philippe Deléham, Nov 11 2009

See A118340 for definition of pendular triangles and pendular sums.

The last five rows in the example section appear in the table on p. 8 of Getzler. Cf. also A173075. - Tom Copeland, Jan 22 2020

			Triangle begins:
  1;
  1,  0;
  1,  1,  0;
  1,  2,  1,  0;
  1,  3,  3,  1,  0;
  1,  4,  7,  4,  1,  0;
  1,  5, 12, 12,  5,  1,  0; ...

G. C. Greubel, Rows n = 0..100 of the triangle, flattened
E. Getzler, The semi-classical approximation for modular operads, arXiv:alg-geom/9612005, 1996.

Cf. this sequence (p=0), A118340 (p=1), A118345 (p=2), A118350 (p=3).

Cf. A001764, A006013, A006629, A093951, A102893, A173075.

function T(n,k,p)
  if k lt 0 or n lt k then return 0;
  elif k eq 0 then return 1;
  elif k eq n then return 0;
  elif n gt 2*k then return T(n,n-k,p) + T(n-1,k,p);
  else return T(n,n-k-1,p) + p*T(n-1,k,p);
  end if;
  return T;
end function;
[T(n,k,0): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 17 2021

T[n_, k_, p_]:= T[n,k,p] = If[nG. C. Greubel, Feb 17 2021 *)

{T(n,k)=if(k==0,1,if(n==k,0,if(n>2*k,binomial(n+k+1,k)*(n-2*k+1)/(n+k+1),T(n,n-1-k))))} \\ Paul D. Hanna, Nov 12 2009

@CachedFunction
def T(n, k, p):
    if (k<0 or n2*k): return T(n,n-k,p) + T(n-1,k,p)
    else: return T(n, n-k-1, p) + p*T(n-1, k, p)
flatten([[T(n, k, 0) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 17 2021

T(2n+m) = [A001764^(m+1)](n), i.e., the m-th lower semi-diagonal forms the self-convolution (m+1)-power of A001764.
If n > 2k, T(n,k) = binomial(n+k+1,k)*(n-2k+1)/(n+k+1), else T(n,k) = T(n,n-1-k), with conditions: T(n,0)=1 for n>=0 and T(n,n)=0 for n > 0. - Paul D. Hanna, Nov 12 2009
Sum_{k=0..n} T(n, k, p=0) = A093951(n). - G. C. Greubel, Feb 17 2021

A109971 Inverse of Riordan array (1,x(1-x)^2), A109970.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 7, 4, 1, 0, 30, 18, 6, 1, 0, 143, 88, 33, 8, 1, 0, 728, 455, 182, 52, 10, 1, 0, 3876, 2448, 1020, 320, 75, 12, 1, 0, 21318, 13566, 5814, 1938, 510, 102, 14, 1, 0, 120175, 76912, 33649, 11704, 3325, 760, 133, 16, 1, 0, 690690, 444015, 197340

Offset: 0

Paul Barry, Jul 06 2005

Row sums are A001764. Diagonal sums are A109972. Second column is A006013. Third column is A006629.

			Rows begin
1;
0,1;
0,2,1;
0,7,4,1;
0,30,18,6,1;
0,143,88,33,8,1;
Production array begins
0, 1
0, 2, 1
0, 3, 2, 1
0, 4, 3, 2, 1
0, 5, 4, 3, 2, 1
0, 6, 5, 4, 3, 2, 1,
0, 7, 6, 5, 4, 3, 2, 1
0, 8, 7, 6, 5, 4, 3, 2, 1
0, 9, 8, 7, 6, 5, 4, 3, 2, 1
... - _Philippe Deléham_, Mar 05 2013

Naiomi Cameron, J. E. McLeod, Returns and Hills on Generalized Dyck Paths, Journal of Integer Sequences, Vol. 19, 2016, #16.6.1.
W.-j. Woan, The Lagrange inversion formula and divisibility properties, JIS 10 (2007) 07.7.8, example 4.

Essentially the same as A092276.

Number triangle T(0, 0)=1, T(0, k)=0, k>0, T(n, k)=(k/n)*binomial(3n-k-1, n-k) otherwise; Riordan array (1, f) where f(1-f)^2=x.
T(n, k)=sum{j=0..n, ((3j+1)/(2n+j+1))(-1)^(j-k)*C(3n, 2n+j)C(j, k)}; - Paul Barry, Oct 07 2005
T(n,k)=binomial(3n-k,n-k)*2k/(3n-k). (Paul Barry, May 18 2006)

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).

A093951 Sum of integers generated by n-1 substitutions, starting with 1, k -> k+1, k-1, .., 1.

Original entry on oeis.org

1, 2, 4, 8, 17, 36, 80, 176, 403, 910, 2128, 4896, 11628, 27132, 65208, 153824, 373175, 888030, 2170740, 5202600, 12797265, 30853680, 76292736, 184863168, 459162452, 1117370696, 2786017120, 6804995008, 17024247304, 41717833740, 104673837384

Offset: 1

Wouter Meeussen, Apr 18 2004

nonn

Substitutions 1 -> {2}, 2 -> {3,1}, 3 -> {4,2}, 4 -> {5,3,1}, 5 -> {6,4,2}, 6 -> {7,5,3,1}, 7 -> {8,6,4,2}, etc. The function f(n) gives determinant of (I_{n} - x * A(n)) where I_{n} is the identity matrix and A(n) = 0 if j > i + 1 otherwise (i+j) mod 2, for i = 1, ..., n and j = 1, ..., n, and can be written in terms of Dickson polynomials as g(w) = x*D_(w-1)(1+x, x*(1+x)) + (1-2*x)*E_(w-1)(1+x, x*(1+x)). - Francisco Salinas (franciscodesalinas(AT)hotmail.com), Apr 13 2004
Count of integers is A047749.
Sum of integers with substitution starting from 0 is A084081.

			GF(12) = (1 + 2*x - 7*x^2 - 14*x^3 + 9*x^4 + 20*x^5 + 2*x^6 - 2*x^7 + 2*x^11)/(1 - 11*x^2 + 36*x^4 - 35*x^6 + 5*x^8) produces a(1) to a(12).
a(4)=8 since 4-1 = 3 substitutions on 1 produce 1 -> 2 -> 3+1 -> 4 + 2 + 2 = 8.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A006629, A047749, A056096, A084081.

function A093951(n)
  if (n mod 2) eq 0 then return 8*Binomial(Floor(3*n/2), Floor((n-2)/2))/(n+2);
  else return 6*Binomial(Floor((3*n+1)/2), Floor((n-1)/2))/(n+2) - 2*Binomial(Floor((3*n-1)/2), Floor((n-1)/2))/(n+1);
  end if; return A093951;
end function;
[A093951(n): n in [1..40]]; // G. C. Greubel, Oct 17 2022

Plus@@@Flatten/@NestList[ #/.k_Integer:>Range[k+1, 1, -2]&, {1}, 8];(*or for n>16 *); f[1]=1; f[2]=1-x^2; f[3]=1-2x^2; f[n_]:=f[n]=Expand[f[n-1]-x^2 f[n-3]]; g[1]=1; g[2]=1+2x; g[3]=1+2x+2x^2; g[n_]:=g[n]=Expand[g[n-1] -x^2 g[n-3]+2 x^(n-1)]; GF[n_]:=g[n]/f[n]; CoefficientList[Series[GF[36], {x, 0, 36}], x]

{a(n)=if(n%2==0,4*binomial(3*n/2,n/2-1)/(n/2+1), 6*binomial(3*(n\2)+2, n\2)/(2*(n\2)+3) - binomial(3*(n\2)+1,n\2)/(n\2+1))} \\ Paul D. Hanna, Apr 24 2006

def A093951(n):
    if (n%2==0): return 8*binomial(3*n/2, (n-2)/2)/(n+2)
    else: return 6*binomial((3*n+1)/2, (n-1)/2)/(n+2) - 2*binomial((3*n-1)/2, (n-1)/2)/(n+1)
[A093951(n) for n in range(1,40)] # G. C. Greubel, Oct 17 2022

a(n) = [x^n] GF(n) with GF(n) = g(n)/f(n) and f(1)=1, f(2)=1-x^2, f(3)=1-2*x^2, f(n) = f(n-1) - x^2*f(n-3) and g(1)=1, g(2)=1+2*x, g(3)=1+2*x+2*x^2, g(n) = g(n-1) - x^2*g(n-3) + 2*x^(n-1).
From Paul D. Hanna, Apr 24 2006: (Start)
a(2*n) = 4*binomial(3*n, n-1)/(n+1) = 2*A006629(n-1).
a(2*n+1) = 6*binomial(3*n+2, n)/(2*n+3) - binomial(3*n+1, n)/(n+1) = A056096(n+3). (End)

A102593 Triangle read by rows: T(n,k) is the number of noncrossing trees with n edges in which the maximum number of contiguous border edges starting from the root in counterclockwise direction is equal to k.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 5, 4, 2, 1, 25, 18, 8, 3, 1, 130, 88, 37, 13, 4, 1, 700, 455, 185, 63, 19, 5, 1, 3876, 2448, 973, 325, 97, 26, 6, 1, 21945, 13566, 5304, 1748, 518, 140, 34, 7, 1, 126500, 76912, 29697, 9690, 2856, 775, 193, 43, 8, 1, 740025, 444015, 169763, 54967

Offset: 0

Emeric Deutsch, Jan 22 2005

Row n has n+1 terms. Row sums yield the ternary numbers (A001764). T(n,0)=A102893(n).

			T(2,0) = T(2,1) = T(2,2) = 1 because in _\, /\ and /_ the maximum number of contiguous border edges starting from the root in counterclockwise direction is 0,1 and 2, respectively.
Triangle starts:
    1;
    0,   1;
    1,   1,   1;
    5,   4,   2,  1;
   25,  18,   8,  3,  1;
  130,  88,  37, 13,  4, 1;
  700, 455, 185, 63, 19, 5, 1;
  ...

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, A102893, A006629.

T:=proc(n,k) if n=0 and k=0 then 1 elif n=1 and k=1 then 1 elif k<=n then (k+1)*binomial(3*n-2*k,n-k)/(2*n-k+1)-(k+2)*binomial(3*n-2*k-2,n-k-1)/(2*n-k) else 0 fi end: for n from 0 to 10 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

T[n_ /; n>1, k_] /; 0 <= k <= n := (k + 1) Binomial[3n - 2k, n - k]/(2n - k + 1) - (k + 2) Binomial[3n - 2k - 2, n - k - 1]/(2n - k); T[1, 1] = T[0, 0] = 1; T[, ] = 0;
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jul 06 2018 *)

T(n,k) = {if(n==0, k==0, if(k<=n, (k+1)*binomial(3*n-2*k, n-k)/(2*n-k+1)-(k+2)*binomial(3*n-2*k-2, n-k-1)/(2*n-k)))} \\ Andrew Howroyd, Nov 06 2017

T(n, k) = (k+1)binomial(3n-2k, n-k)/(2n-k+1) - (k+2)binomial(3n-2k-2, n-k-1)/(2n-k) if n > 1, 0 <= k <= n; T(1, 1)=1; T(0, 0)=1; T(n, k)=0 if k > n.

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

A109956 Inverse of Riordan array (1/(1-x), x/(1-x)^3), A109955.

Original entry on oeis.org

1, -1, 1, 3, -4, 1, -12, 18, -7, 1, 55, -88, 42, -10, 1, -273, 455, -245, 75, -13, 1, 1428, -2448, 1428, -510, 117, -16, 1, -7752, 13566, -8379, 3325, -910, 168, -19, 1, 43263, -76912, 49588, -21252, 6578, -1472, 228, -22, 1, -246675, 444015, -296010, 134550, -45630, 11700, -2223, 297, -25, 1

Offset: 0

Paul Barry, Jul 06 2005

Riordan array (g,f) where f/(1-f)^3=x and g=1-f.
First column is (-1)^n*binomial(3n,n)/(2n+1), a signed version of A001764.
Second column is a signed version of A006629.
Diagonal sums are A109957.

			Triangle begins:
      1;
     -1,   1;
      3,  -4,    1;
    -12,  18,   -7,   1;
     55, -88,   42, -10,   1;
   -273, 455, -245,  75, -13, 1;
   ...

Paul Barry, d-orthogonal polynomials, Fuss-Catalan matrices and lattice paths, arXiv:2505.16718 [math.CO], 2025. See p. 13.
Paul Drube, Generalized Path Pairs and Fuss-Catalan Triangles, arXiv:2007.01892 [math.CO], 2020. See Figure 4 p. 8 (up to signs).

Cf. A001764, A109955, A109957, A321620.

# Function RiordanSquare defined in A321620.
tt := sin(arcsin(3*sqrt(x*3/4))/3)/sqrt(x*3/4): R := RiordanSquare(tt, 11):
seq(seq(LinearAlgebra:-Row(R,n)[k]*(-1)^(n+k), k=1..n), n=1..11); # Peter Luschny, Nov 27 2018

T[n_, k_] := (-1)^(n - k)((3k + 1)/(2n + k + 1)) Binomial[3n, n - k];
Table[T[n, k], {n, 0, 9}, {k, 0, n}] (* Jean-François Alcover, Jun 13 2019 *)

tabl(nn) = {my(m = matrix(nn, nn, n, k, if (nMichel Marcus, Nov 20 2015

Number triangle T(n, k) = (-1)^(n-k)*((3k+1)/(2n+k+1))*binomial(3n, n-k).
From Werner Schulte, Oct 27 2015: (Start)
If u(m,n) = (-1)^n*(Sum_{k=0..n} T(n,k)*((m+1)*k+1)) and v(m,n) = (-1)^n*(Sum_{k=0..n} (-1)^k*T(n,k)*m^k) and D(x) is the g.f. of A001764 then P(m,x) = Sum_{n>=0} u(m,n)*x^n = 1-(m+1)*x*D(x)^2 and Q(m,x) = Sum_{n>=0} v(m,n)*x^n = 1/P(m,x).
If G(k,x) is the g.f. of column k (k>=0) then G(k,x) = G(0,x)^(3*k+1). (End)

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)).

cp's OEIS Frontend

A110616 A convolution triangle of numbers based on A001764.

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A167763 Pendular triangle (p=0), read by rows, where row n is formed from row n-1 by the recurrence: if n > 2k, T(n,k) = T(n,n-k) + T(n-1,k), otherwise T(n,k) = T(n,n-1-k) + p*T(n-1,k), for n >= k <= 0, with T(n,0) = 1 and T(n,n) = 0^n.

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A109971 Inverse of Riordan array (1,x(1-x)^2), A109970.

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

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A093951 Sum of integers generated by n-1 substitutions, starting with 1, k -> k+1, k-1, .., 1.

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

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