A235340 -id:A235340 - cp's OEIS frontend

A118971 a(n) = binomial(5*n+3,n)/(n+1).

1, 4, 26, 204, 1771, 16380, 158224, 1577532, 16112057, 167710664, 1772645420, 18974357220, 205263418941, 2240623268512, 24648785802336, 272994644359580, 3041495503591365, 34064252968167180, 383302465665133014

Offset: 0

Paul Barry, May 07 2006

A quadrisection of A118968.
For n >= 1, a(n-1) is the number of lattice paths from (0,0) to (4n,n) using only the steps (1,0) and (0,1) and which stay strictly below the line y = x/4 except at the path's endpoints. -  Lucas A. Brown, Aug 21 2020
This is instance k = 4 of the family {c(k, n+1)}A130564.%20-%20_Wolfdieter%20Lang">{n>=0} given in a comment in A130564. - _Wolfdieter Lang, Feb 04 2024

Michael De Vlieger, Table of n, a(n) for n = 0..924
Paul Barry, Characterizations of the Borel triangle and Borel polynomials, arXiv:2001.08799 [math.CO], 2020.
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
J. Sawada, J. Sears, A. Trautrim, and A. Williams, Demystifying our Grandparent's De Bruijn Sequences with Concatenation Trees, arXiv:2308.12405 [math.CO], 2023.

Cf. A000108, A006013, A006632, A130564, A130565, A234466, A234513, A234573, A235340 (members of the same family).

Table[4*Binomial[5n+3,n]/(4n+4),{n,0,30}] (* Harvey P. Dale, Apr 09 2012 *)

G.f.: If the inverse series of y*(1-y)^4 is G(x) then A(x)=G(x)/x.
D-finite with recurrence 8*(4*n+1)*(2*n+1)*(4*n+3)*(n+1)*a(n) -5*(5*n+1)*(5*n+2)*(5*n+3)*(5*n-1)*a(n-1)=0. - R. J. Mathar, Nov 26 2012
a(n) = (4/5)*binomial(5*(n+1),n+1)/(5*(n+1)-1). - Bruno Berselli, Jan 17 2014
E.g.f.: 4F4(4/5,6/5,7/5,8/5; 5/4,3/2,7/4,2; 3125*x/256). - Ilya Gutkovskiy, Jan 23 2018
G.f.: 5F4([4,5,6,7,8]/5, [5,6,7,8]/4; (5^5/4^4)*x) = (4/(5*x))*(1 - 4F3([-1,1,2,3]/5, [1,2,3]/4; (5^5/4^4)*x)). - Wolfdieter Lang, Feb 15 2024

A130564 Member k=5 of a family of generalized Catalan numbers.

Original entry on oeis.org

1, 5, 40, 385, 4095, 46376, 548340, 6690585, 83615350, 1064887395, 13770292256, 180320238280, 2386316821325, 31864803599700, 428798445360120, 5809228810425801, 79168272296871450, 1084567603590147950

Offset: 1

Wolfdieter Lang, Jul 13 2007

The generalized Catalan numbers C(k,n):= binomial(k*n+1,n)/(k*n+1) become for negative k=-|k|, with |k|>=2, ((-1)^(n-1))*binomial((|k|+1)*n-2,n)/(|k|*n-1), n>=0.
The family c(k,n):=binomial((k+1)*n-2,n)/(k*n-1), n>=1, has the members A000108, A006013, A006632, A118971 for k=1,2,3,4, respectively (but the offset there is 0).
The members of the C(k,n) family for positive k are: A000012 (powers of 1), A000108, A001764, A002293, A002294, A002295, A002296, A007556, A062994, for k=1..9.
The ordinary generating functions for the k-family {c(k, n+1)}{n>=0}, k >= 1, are G(k, x) = hypergeometric(Aseq(k+1), Bseq(k), ((k+1)^(k+1)/k^k)*x), with Aseq(k+1) = [a(k)_1,..., a(k){k+1}], where a(k)j = (2*k - (j-1))/(k+1), and Bseq(k) = [b(k)_1, ..., b(k)_k], where b(k)_j = (2*k - (j-1))/k. The e.g.f. has an extra 1 in the B-section, which leads to a cancellation with the A-section 1 term. Thanks to Dixon J. Jones for asking for the general formulas. - _Wolfdieter Lang, Feb 04 2024
The o.g.f. of {C(k, n)}{n>=0} is in the Graham-Knuth-Patashnik book denoted as B_k(z) on pp. 200, 349 (2nd ed. 1994, pp. 200, 363). - _Wolfdieter Lang, Mar 07 2024

Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics, Addison-Wesley, Reading, MA, 2nd ed. 1994, pp. 200, 363.

Michael De Vlieger, Table of n, a(n) for n = 1..856
K. Kobayashi, H. Morita and M. Hoshi, Coding of ordered trees, Proceedings, IEEE International Symposium on Information Theory, ISIT 2000, Sorrento, Italy, Jun 25 2000.
Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

Cf. A000012, A000108, A001764, A002293, A002294, A002295, A002296, A006013, A062994, A006632, A007556, A118971, A130565, A234466, A234513, A234573, A235340.

Rest@ CoefficientList[InverseSeries[Series[y (1 - y)^5, {y, 0, 18}], x], x] (* Michael De Vlieger, Oct 13 2019 *)

a(n) = binomial((k+1)*n-2,n)/(k*n-1), with k=5.
G.f.: inverse series of y*(1-y)^5.
a(n) = (5/6)*binomial(6*n,n)/(6*n-1). [Bruno Berselli, Jan 17 2014]
From Wolfdieter Lang, Feb 06 2020: (Start)
G.f.: (5/6)*(1 - hypergeom([-1, 1, 2, 3, 4]/6, [1, 2, 3, 4]/5,(6^6/5^5)*x)).
E.g.f.: (5/6)*(1 - hypergeom([-1, 1, 2, 3, 4]/6, [1, 2, 3, 4, 5]/5,(6^6/5^5)*x)). (End)
D-finite with recurrence 5*n*(5*n-4)*(5*n-3)*(5*n-2)*(5*n-1)*a(n) -72*(6*n-7)*(3*n-1)*(2*n-1)*(3*n-2)*(6*n-5)*a(n-1)=0. - R. J. Mathar, May 07 2021

A235339 a(n) = 9binomial(11n+9,n)/(11*n+9).

Original entry on oeis.org

1, 9, 135, 2460, 49725, 1072197, 24163146, 562311720, 13409091540, 325949656825, 8046743477058, 201198155083200, 5084704634041305, 129673310477725350, 3332952595603387800, 86250038091202771344, 2245329811618166111985

Offset: 0

Tim Fulford, Jan 06 2014

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p = 11, 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.
Wikipedia, Fuss-Catalan number

Cf. A230388, A234868, A234869, A234870, A234871, A234872, A234873, A235338, A235340, A069271, A118970, A212073, A230388, A233834, A234465, A234510, A234571.

[9*Binomial(11*n+9, n)/(11*n+9): n in [0..30]];

Table[9 Binomial[11 n + 9, n]/(11 n + 9), {n, 0, 30}]

PARI
```
a(n) = 9*binomial(11*n+9,n)/(11*n+9);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(11/9))^9+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, here p = 11, r = 9.

O.g.f. A(x) = 1/x * series reversion (x/C(x)^9), where C(x) is the o.g.f. for the Catalan numbers A000108. A(x)^(1/9) is the o.g.f. for A230388. - Peter Bala, Oct 14 2015

A235338 a(n) = 8binomial(11n+8,n)/(11*n+8).

Original entry on oeis.org

1, 8, 116, 2080, 41650, 892552, 20027112, 464550336, 11050084695, 268070745800, 6607118937848, 164979021222400, 4164615224071926, 106105019316578800, 2724883054841727200, 70462458864489354624, 1833143662625459289495

Offset: 0

Tim Fulford, Jan 06 2014

nonn

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=11, r=8.

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 [broken link]
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.

Cf. A230388, A234868, A234869, A234870, A234871, A234872, A234873, A235339, A235340.

[8*Binomial(11*n+8, n)/(11*n+8): n in [0..30]];

Table[8 Binomial[11 n + 8, n]/(11 n + 8), {n, 0, 30}]

PARI
```
a(n) = 8*binomial(11*n+8, n)/(11*n+8);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(11/8))^8+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, here p=11, r=8.

A386558 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = binomial((k+1)*n+k-1,n)/(n+1).

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 3, 7, 5, 0, 1, 4, 15, 30, 14, 0, 1, 5, 26, 91, 143, 42, 0, 1, 6, 40, 204, 612, 728, 132, 0, 1, 7, 57, 385, 1771, 4389, 3876, 429, 0, 1, 8, 77, 650, 4095, 16380, 32890, 21318, 1430, 0, 1, 9, 100, 1015, 8184, 46376, 158224, 254475, 120175, 4862, 0

Offset: 0

Seiichi Manyama, Jul 26 2025

			Square array begins:
  1,   1,    1,     1,      1,      1,       1, ...
  0,   1,    2,     3,      4,      5,       6, ...
  0,   2,    7,    15,     26,     40,      57, ...
  0,   5,   30,    91,    204,    385,     650, ...
  0,  14,  143,   612,   1771,   4095,    8184, ...
  0,  42,  728,  4389,  16380,  46376,  109668, ...
  0, 132, 3876, 32890, 158224, 548340, 1533939, ...

Wikipedia, Fuss-Catalan number

Columns k=0..10 give A000007, A000108, A006013, A006632, A118971, A130564(n+1), A130565(n+1), A234466, A234513, A234573, A235340.
Main diagonal gives A177784(n+1).
Cf. A162382.

a(n, k) = binomial((k+1)*n+k-1, n)/(n+1);

For k > 0, A(n,k) = r * binomial(n*p+r,n)/(n*p+r), the Fuss-Catalan number with p=k+1 and r=k.

G.f. of column k: (1/x) Series_Reverion( x*(1-x)^k ).

cp's OEIS Frontend

A118971 a(n) = binomial(5*n+3,n)/(n+1).

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A130564 Member k=5 of a family of generalized Catalan numbers.

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A235339 a(n) = 9*binomial(11*n+9,n)/(11*n+9).

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A235338 a(n) = 8*binomial(11*n+8,n)/(11*n+8).

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Magma

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A386558 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals: A(n,k) = binomial((k+1)*n+k-1,n)/(n+1).

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A235339 a(n) = 9binomial(11n+9,n)/(11*n+9).

A235338 a(n) = 8binomial(11n+8,n)/(11*n+8).