A062993 -id:A062993 - cp's OEIS frontend

A230388 a(n) = binomial(11n+1,n)/(11n+1).

1, 1, 11, 176, 3311, 68211, 1489488, 33870540, 793542167, 19022318084, 464333035881, 11502251937176, 288417894029200, 7306488667126803, 186719056586568660, 4807757550367267056, 124609430032127192295, 3248403420844673986345

Offset: 0

Tim Fulford, Jan 01 2014

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=11, r=1. Interesting property when r=1, a(n+1,p,1) = a(n,p,p) for n>=0.

This is also instance k = 10 of the generalized Catalan family {C(k, n)}A130564.%20-%20_Wolfdieter%20Lang">{n>=0} given in a comment of A130564. - _Wolfdieter Lang, Feb 05 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906 [math.CO], 2008.
J-C. Aval, Multivariate Fuss-Catalan Numbers, 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.
J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962, 2014
Wikipedia, Fuss-Catalan number

Cf. A234868-A234873, A130564.

[Binomial(11*n+1,n)/(11*n+1): n in [0..30]]; // Vincenzo Librandi, Jan 01 2014

seq(binomial(11*k+1,k)/(11*k+1),k=0..30); # Robert FERREOL, Apr 01 2015
n:=30:G:=series(RootOf(g = 1+x*g^11, g),x=0,n+1):seq(coeff(G,x,k),k=0..n); # Robert FERREOL, Apr 01 2015

Table[Binomial[11 n + 1, n]/(11 n + 1), {n, 0, 30}] (* Vincenzo Librandi, Jan 01 2014 *)

PARI
```
a(n) = binomial(11*n+1,n)/(11*n+1);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(11))+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, with p=11, r=1.
From Robert FERREOL, Apr 01 2015: (Start)
a(n) = binomial(11*n,n)/(10*n+1) = A062993(n+9, 9).
a(0) = 1; a(n) = Sum_{i1+i2+..i11=n-1} a(i1)*a(i2)*...*a(i11) for n>=1.
(End)
O.g.f.: hypergeometric([1,...,10]/11,[2,...,9,11]/10,(11^11/10^10)*x). For the e.g.f. put an extra  1 = 10/10 into the second part. - Wolfdieter Lang, Feb 05 2024
a(n) ~ (11^11/10^10)^n*sqrt(11/(2*Pi*(10*n)^3)). - Robert A. Russell, Jul 15 2024
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^21). - Seiichi Manyama, Jun 16 2025

A355262 Array of Fuss-Catalan numbers read by ascending antidiagonals, A(n, k) = binomial(kn + 1, k)/(kn + 1).

Original entry on oeis.org

1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 3, 5, 1, 0, 1, 1, 4, 12, 14, 1, 0, 1, 1, 5, 22, 55, 42, 1, 0, 1, 1, 6, 35, 140, 273, 132, 1, 0, 1, 1, 7, 51, 285, 969, 1428, 429, 1, 0, 1, 1, 8, 70, 506, 2530, 7084, 7752, 1430, 1, 0

Offset: 0

Peter Luschny, Jun 26 2022

An alternative definition is: the Fuss-Catalan sequences (A(n, k), k >= 0 ) are the main diagonals of the Fuss-Catalan triangles of order n - 1. See A355173 for the definition of a Fuss-Catalan triangle.

			Array A(n, k) begins:
[0] 1, 1, 0,   0,    0,     0,      0,       0,         0, ...  A019590
[1] 1, 1, 1,   1,    1,     1,      1,       1,         1, ...  A000012
[2] 1, 1, 2,   5,   14,    42,    132,     429,      1430, ...  A000108
[3] 1, 1, 3,  12,   55,   273,   1428,    7752,     43263, ...  A001764
[4] 1, 1, 4,  22,  140,   969,   7084,   53820,    420732, ...  A002293
[5] 1, 1, 5,  35,  285,  2530,  23751,  231880,   2330445, ...  A002294
[6] 1, 1, 6,  51,  506,  5481,  62832,  749398,   9203634, ...  A002295
[7] 1, 1, 7,  70,  819, 10472, 141778, 1997688,  28989675, ...  A002296
[8] 1, 1, 8,  92, 1240, 18278, 285384, 4638348,  77652024, ...  A007556
[9] 1, 1, 9, 117, 1785, 29799, 527085, 9706503, 184138713, ...  A062994

N. I. Fuss, Solutio quaestionis, quot modis polygonum n laterum in polygona m laterum, per diagonales resolvi queat, Nova Acta Academiae Scientiarum Imperialis Petropolitanae, vol.9 (1791), 243-251.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 1990, (Eqs. 5.70, 7.66, and sec. 7.5, example 5).

Per Alexandersson, Frether Getachew Kebede, Samuel Asefa Fufa, and Dun Qiu, Pattern-Avoidance and Fuss-Catalan Numbers, J. Int. Seq. (2023) Vol. 26, Art. 23.4.2.
Joerg Arndt, Matters Computational (The Fxtbook), pp. 337-338.
Jean-Luc Baril, Mireille Bousquet-Mélou, Sergey Kirgizov, and Mehdi Naima, The ascent lattice on Dyck paths, arXiv:2409.15982 [math.CO], 2024. See p. 6.
Jean-Christophe Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906v1, Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers, Chapter 7.
D. E. Knuth, Convolution polynomials, arXiv:math/9207221 [math.CA]; Mathematica J. 2.1 (1992), no. 4, 67-78.
Donald Knuth's 20th Annual Christmas Tree Lecture, (3/2)-ary Trees, Stanford Online, Video 2014.
Wojciech Młotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Math. 15:939-955, (2010).
Wikipedia, Fuss-Catalan number.

Cf. A091144 (main diagonal), A019590, A000012, A000108, A001764, A002293, A002294, A002295, A002296, A007556, A062994.
Variants: A062993, A070914.
Fuss-Catalan triangles: A123110 (order 0), A355173 (order 1), A355172 (order 2), A355174 (order 3).

A := (n, k) -> binomial(k*n + 1, k)/(k*n + 1):
for n from 0 to 9 do seq(A(n, k), k = 0..8) od;

(* See the Knuth references. In the christmas lecture Knuth has fun calculating the Fuss-Catalan development of Pi and i. *)
B[t_, n_] := Sum[Binomial[t k+1, k] z^k / (t k+1), {k, 0, n-1}] + O[z]^n
Table[CoefficientList[B[n, 9], z], {n, 0, 9}] // TableForm

A(n, k) = (1/k!) * Product_{j=1..k-1} (k*n + 1 - j).
A(n, k) = (binomial(k*n, k) + binomial(k*n, k-1)) / (k*n + 1).
Let B(t, z) = Sum_{k>=0} binomial(k*t + 1, k)*z^k / (k*t + 1), then
A(n, k) = [z^k] B(n, z).

A091144 a(n) = binomial(n^2, n)/(1+(n-1)*n).

Original entry on oeis.org

1, 1, 2, 12, 140, 2530, 62832, 1997688, 77652024, 3573805950, 190223180840, 11502251937176, 779092434772236, 58448142042957576, 4811642166029230560, 431306008583779517040, 41820546066482630185200

Offset: 0

Paul Barry, Dec 22 2003

Diagonal of array T(n,k) = binomial(kn,n)/(1+(k-1)n).
Number of paths up and left from (0,0) to (n^2-n,n) where x/y <= n-1 for all intermediate points. - Henry Bottomley, Dec 25 2003
Empirical: In the ring of symmetric functions over the fraction field Q(q, t), letting s(1^n) denote the Schur function indexed by (1^n), a(n) is equal to the coefficient of s(n) in nabla^(n)s(1^n) with q=t=1, where nabla denotes the "nabla operator" on symmetric functions, and s(n) denotes the Schur function indexed by the integer partition (n) of n. - John M. Campbell, Apr 06 2018

Vincenzo Librandi, Table of n, a(n) for n = 0..200
D. Merlini, R. Sprugnoli and M. C. Verri, The tennis ball problem, J. Combin. Theory, A 99 (2002), 307-344.

Cf. A002061, A014062, A062993, A070914, A071201, A071202.

List([0..20],n->Binomial(n^2,n)/(1+(n-1)*n)); # Muniru A Asiru, Apr 08 2018

[Binomial(n^2, n)/(1+(n-1)*n): n in [0..20]]; // Vincenzo Librandi, Apr 07 2018

A091144 := proc(n)
    binomial(n^2,n)/(1+n*(n-1)) ;
end proc: # R. J. Mathar, Feb 14 2015

Table[Binomial[n^2, n] / (n (n - 1) + 1), {n, 0, 20}] (* Vincenzo Librandi, Apr 07 2018 *)

a(n) = binomial(n^2, n)/(n*(n-1)+1); \\ Altug Alkan, Apr 06 2018

From Henry Bottomley, Dec 25 2003: (Start)
a(n) = A014062(n)/A002061(n);
a(n) = A062993(n-2, n);
a(n) = A070914(n, n-1);
a(n) = A071201(n, n^2-n);
a(n) = A071201(n, n^2-n+1);
a(n) = A071202(n, n^2-n+1). (End)

A062747 Row sums of (unsigned) staircase array A062746.

Original entry on oeis.org

1, 7, 89, 1447, 26713, 532391, 11165785, 242851751, 5427716185, 123901026215, 2876525797465, 67710590623655, 1612262780199001, 38764533106581415, 939825790848884825, 22950405085586497447

Offset: 0

Wolfdieter Lang, Jul 12 2001

a(n)=N(3; k, x=-1), with the polynomials N(3; k, x) from the staircase array A062746.
a(n) = 2*( Sum_{j = 0..n} (-1)^j*C(3; n-j)*4^(n-j) ) - (-1)^n with C(3; n) := A001764(n) = A062993(n+1, 1) (a Pfaff-Fuss or 3-Raney sequence).
G.f.: (2*c(3; 4*x)-1)/(1+x) with c(3; x)= RootOf(x*A001764%20%5Bformula%20for%20a(n)%20and%20g.f.%20corrected%20by%20_Peter%20Bala">Z^3-_Z +1), the g.f. of A001764 [formula for a(n) and g.f. corrected by _Peter Bala, Mar 26 2020].
Conjectural recurrence: n*(2*n+1)*a(n) = (4*n-3)*(13*n-4)*a(n-1) + 6*(3*n-1)*(3*n-2)*a(n-2) with a(0) = 1, a(1) = 7. - Peter Bala, Mar 25 2020

A062752 Row sums of unsigned N(4) staircase array A062751.

Original entry on oeis.org

1, 15, 497, 22031, 1124849, 62379535, 3651676657, 222085764623, 13895337519601, 888654458770959, 57831897893972465, 3817410543738148367, 254970980461934291441, 17200148833928765494799

Offset: 0

Wolfdieter Lang, Jul 12 2001

a(n)=N(4; n, -1) with the row polynomials N(4; n, x) defined in A062751.

a(n)=sum(((-1)^(n-j))*2^(3*j+1)*A002293(j), j=1..n)+(-1)^n, with A002293(j)= A062993(j+2, 2)= binomial(4*j, j)/(3*j+1).

A062987 Row sums of unsigned N(5) staircase array A062986.

Original entry on oeis.org

1, 31, 2529, 284191, 37071329, 5268723231, 791682591201, 123697944483359, 19894672175770081, 3271817054307112479, 547678880100062177761, 93006445178165754746399, 15983911852747899752786401

Offset: 0

Wolfdieter Lang, Jul 12 2001

Cf. A002294, A062986, A062993.

a(n) = N(5; n, -1) with polynomials N(5; n, x) defined in A062986.

a(n) = Sum(((-1)^(n-j))*2^(4*j+1)*A002294(j), j=1..n)+(-1)^n, with A002294(j) = A062993(j+3, 3) = binomial(5*j, j)/(4*j+1).

A137211 Generalized or s-Catalan numbers.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 5, 12, 22, 1, 14, 55, 140, 285, 1, 42, 273, 969, 2530, 5481, 1, 132, 1428, 7084, 23751, 62832, 141778, 1, 429, 7752, 53820, 231880, 749398, 1997688, 4638348, 1, 1430, 43263, 420732, 2330445, 9203634, 28989675, 77652024

Offset: 1

Roger L. Bagula, Mar 05 2008

nonn

From R. J. Mathar, May 04 2008: (Start)
This is a triangular section of Stanica's array of s-Catalan numbers, with rows A000108, A001764, A002293-A002296, A007556, A062994, A059968,... read along diagonals in A062993 and A070914:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796, ...
1, 1, 3, 12, 55, 273, 1428, 7752, 43263, 246675, 1430715, ...
1, 1, 4, 22, 140, 969, 7084, 53820, 420732, 3362260, 27343888, ...
1, 1, 5, 35, 285, 2530, 23751, 231880, 2330445, 23950355, 250543370, ...
1, 1, 6, 51, 506, 5481, 62832, 749398, 9203634, 115607310, 1478314266, ...
1, 1, 7, 70, 819, 10472, 141778, 1997688, 28989675, 430321633, 6503352856, ...
1, 1, 8, 92, 1240, 18278, 285384, 4638348, 77652024, 1329890705, 23190029720, ...
1, 1, 9, 117, 1785, 29799, 527085, 9706503, 184138713, 3573805950, 70625252863, ...
1, 1, 10, 145, 2470, 46060, 910252, 18730855, 397089550, 8612835715, 190223180840, ...
(End)
The Fuss-Catalan numbers are Cat(d,k)= [1/(k*(d-1)+1)]*binomial(k*d,k) and enumerate the number of (d+1)-gon partitions of a (k*(d-1)+2)-gon (cf. Whieldon and Schuetz link for this interpretation and others), so the (k+1)-th column of Stanica's array enumerates the number of (n+1)-gon partitions of a (k*(n-1)+2)-gon. Cf. A000326 (k=3), A100157 (k=4) and A234043 (k=5). - Tom Copeland, Oct 05 2014

			{1},
{1, 1},
{1, 2, 3},
{1, 5, 12, 22},
{1, 14, 55, 140, 285},
{1, 42, 273, 969, 2530, 5481},
{1, 132, 1428, 7084, 23751, 62832, 141778},
{1, 429, 7752, 53820, 231880, 749398, 1997688, 4638348}

Heinrich Niederhausen, Catalan Traffic at the Beach, Electronic Journal of Combinatorics, Volume 9 (2002), #R33.
A. Regev, The Central Component of a Triangulation, J. Int. Seq. 16 (2013) #13.4.1
Alison Schuetz and Gwyneth Whieldon, Polygonal Dissections and Reversions of Series, arXiv:1401.7194 [math.CO], 2014.
P. Stanica, p^q-Catalan numbers and squarefree binomial coefficients, J. Numb. Theory 100 (2003) 203-216.

t[n_, m_] := Binomial[m*n, n]/((m - 1)*n + 1); a = Table[Table[t[n, m], {m, 1, n + 1}], {n, 0, 10}]; Flatten[a]

T(n,m) = binomial(m*n,n)/((m-1)*n+1).

Edited by N. J. A. Sloane, May 16 2008

cp's OEIS Frontend

A230388 a(n) = binomial(11*n+1,n)/(11*n+1).

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A355262 Array of Fuss-Catalan numbers read by ascending antidiagonals, A(n, k) = binomial(k*n + 1, k)/(k*n + 1).

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A091144 a(n) = binomial(n^2, n)/(1+(n-1)*n).

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A062747 Row sums of (unsigned) staircase array A062746.

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A062752 Row sums of unsigned N(4) staircase array A062751.

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A062987 Row sums of unsigned N(5) staircase array A062986.

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A137211 Generalized or s-Catalan numbers.

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A230388 a(n) = binomial(11n+1,n)/(11n+1).

A355262 Array of Fuss-Catalan numbers read by ascending antidiagonals, A(n, k) = binomial(kn + 1, k)/(kn + 1).