A130565 -id:A130565 - 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

A233834 a(n) = 5binomial(7n+5,n)/(7*n+5).

Original entry on oeis.org

1, 5, 45, 500, 6200, 82251, 1142295, 16398200, 241379325, 3623534200, 55262073757, 853814730600, 13335836817420, 210225027967325, 3340362288091500, 53443628421286320, 860246972339613855, 13921016318025200505, 226352372251889455000, 3696160728052814340000

Offset: 0

Tim Fulford, Dec 16 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p = 7, r = 5.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906v1, 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. A000108, A002296, A233832, A233833, A143547, A130565, A233835, A233907, A233908, A002296, A069271, A118970, A212073, A234465, A234510, A234571, A235339.

[5*Binomial(7*n+5, n)/(7*n+5): n in [0..30]];

Table[5 Binomial[7 n + 5, n]/(7 n + 5), {n, 0, 30}]

PARI
```
a(n) = 5*binomial(7*n+5,n)/(7*n+5);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(7/5))^5+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, where p = 7, r = 5.

O.g.f. A(x) = 1/x * series reversion (x/C(x)^5), where C(x) is the o.g.f. for the Catalan numbers A000108. A(x)^(1/5) is the o.g.f. for A002296. - Peter Bala, Oct 14 2015

A251577 E.g.f.: exp(7xG(x)^6) / G(x)^6 where G(x) = 1 + x*G(x)^7 is the g.f. of A002296.

Original entry on oeis.org

1, 1, 7, 133, 4501, 224497, 14926387, 1245099709, 125177105641, 14743403405857, 1991987858095039, 303781606238806549, 51624122993243471293, 9674836841745014156497, 1982441139367342976694379, 440946185623028320815311053, 105810290178441439797537070033, 27247415403508413760437930799681

Offset: 0

Paul D. Hanna, Dec 06 2014

nonn

			E.g.f.: A(x) = 1 + x + 7*x^2/2! + 133*x^3/3! + 4501*x^4/4! + 224497*x^5/5! +...
such that A(x) = exp(7*x*G(x)^6) / G(x)^6
where G(x) = 1 + x*G(x)^7 is the g.f. of A002296:
G(x) = 1 + x + 7*x^2 + 70*x^3 + 819*x^4 + 10472*x^5 + 141778*x^6 +...
Note that
A'(x) = exp(7*x*G(x)^6) = 1 + 7*x + 133*x^2/2! + 4501*x^3/3! +...
LOGARITHMIC DERIVATIVE.
The logarithm of the e.g.f. begins:
log(A(x)) = x + 6*x^2/2 + 57*x^3/3 + 650*x^4/4 + 8184*x^5/5 + 109668*x^6/6 +...
and so A'(x)/A(x) = G(x)^6.
TABLE OF POWERS OF E.G.F.
Form a table of coefficients of x^k/k! in A(x)^n as follows.
n=1: [1, 1,  7,   133,  4501,  224497,  14926387,  1245099709, ...];
n=2: [1, 2,  16,  308, 10360,  512624,  33845728,  2807075264, ...];
n=3: [1, 3,  27,  531, 17829,  876771,  57529143,  4745597787, ...];
n=4: [1, 4,  40,  808, 27184, 1331008,  86864512,  7129675840, ...];
n=5: [1, 5,  55, 1145, 38725, 1891205, 122869075, 10038831425, ...];
n=6: [1, 6,  72, 1548, 52776, 2575152, 166702752, 13564381824, ...];
n=7: [1, 7,  91, 2023, 69685, 3402679, 219682183, 17810832319, ...];
n=8: [1, 8, 112, 2576, 89824, 4395776, 283295488, 22897384832, ...]; ...
in which the main diagonal begins (see A251587):
[1, 2, 27, 808, 38725, 2575152, 219682183, 22897384832, ...]
and is given by the formula:
[x^n/n!] A(x)^(n+1) = 7^(n-5) * (n+1)^(n-6) * (1296*n^5 + 9720*n^4 + 30555*n^3 + 50665*n^2 + 44621*n + 16807) for n>=0.

Cf. A251587, A251667, A002296, A130565.

Cf. Variants: A243953, A251573, A251574, A251575, A251576, A251578, A251579, A251580.

Flatten[{1,1,Table[Sum[7^k * n!/k! * Binomial[7*n-k-7, n-k] * (k-1)/(n-1),{k,0,n}],{n,2,20}]}] (* Vaclav Kotesovec, Dec 07 2014 *)

{a(n) = local(G=1);for(i=1,n,G=1+x*G^7 +x*O(x^n)); n!*polcoeff(exp(7*x*G^6)/G^6, n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = if(n==0|n==1, 1, sum(k=0, n, 7^k * n!/k! * binomial(7*n-k-7,n-k) * (k-1)/(n-1) ))}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^7 be the g.f. of A002296, then the e.g.f. A(x) of this sequence satisfies:
(1) A'(x)/A(x) = G(x)^6.
(2) A'(x) = exp(7*x*G(x)^6).
(3) A(x) = exp( Integral G(x)^6 dx ).
(4) A(x) = exp( Sum_{n>=1} A130565(n)*x^n/n ), where A130565(n) = binomial(7*n-2,n)/(6*n-1).
(5) A(x) = F(x/A(x)) where F(x) is the e.g.f. of A251587.
(6) A(x) = Sum_{n>=0} A251587(n)*(x/A(x))^n/n! and
(7) [x^n/n!] A(x)^(n+1) = (n+1)*A251587(n),
where A251587(n) = 7^(n-5) * (n+1)^(n-7) * (1296*n^5 + 9720*n^4 + 30555*n^3 + 50665*n^2 + 44621*n + 16807).
a(n) = Sum_{k=0..n} 7^k * n!/k! * binomial(7*n-k-7, n-k) * (k-1)/(n-1) for n>1.
Recurrence: 72*(2*n-3)*(3*n-5)*(3*n-4)*(6*n-11)*(6*n-7)*(2401*n^5 - 32585*n^4 + 178311*n^3 - 492779*n^2 + 689623*n - 392491)*a(n) = 7*(282475249*n^11 - 6658345155*n^10 + 71339412375*n^9 - 458968749330*n^8 + 1971937124661*n^7 - 5947597074909*n^6 + 12867618998885*n^5 - 20002508046570*n^4 + 21938241804255*n^3 - 16207858252075*n^2 + 7281095411817*n - 1512276480000)*a(n-1) - 823543*(2401*n^5 - 20580*n^4 + 71981*n^3 - 129346*n^2 + 120663*n - 47520)*a(n-2). - Vaclav Kotesovec, Dec 07 2014
a(n) ~ 7^(7*(n-1)-1/2) / 6^(6*(n-1)-1/2) * n^(n-2) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014

A233832 a(n) = 2binomial(7n+2,n)/(7*n+2).

Original entry on oeis.org

1, 2, 15, 154, 1827, 23562, 320866, 4540200, 66096459, 983592304, 14894775896, 228784720710, 3555866673450, 55819631671902, 883738853546472, 14094715154157680, 226245021605612955, 3652242142988400570, 59254515909624764575, 965678197027521177200

Offset: 0

Tim Fulford, Dec 16 2013

nonn

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=7, r=2.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906v1, Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers Chapter 7
Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k, arXiv:2204.14023 [math.CO], 2022.
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.
Wikipedia, Fuss-Catalan number
Jun Yan, Lattice paths enumerations weighted by ascent lengths, arXiv:2501.01152 [math.CO], 2025. See p. 7.
Sheng-liang Yang and Mei-yang Jiang, Pattern avoiding problems on the hybrid d-trees, J. Lanzhou Univ. Tech., (China, 2023) Vol. 49, No. 2, 144-150. (in Mandarin)

Cf. A000108, A002296, A233833 - A233835, A143547, A130565, A233907, A233908.

[2*Binomial(7*n+2, n)/(7*n+2): n in [0..30]];

Table[2 Binomial[7 n + 2, n]/(7 n + 2), {n, 0, 30}]

PARI
```
a(n) = 2*binomial(7*n+2,n)/(7*n+2);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(7/2))^2+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=7, r=2.
a(n) = 2*binomial(7n+1, n-1)/n for n>0, a(0)=1. [Bruno Berselli, Jan 19 2014]
From Ilya Gutkovskiy, Sep 14 2018: (Start)
E.g.f.: 6F6(2/7,3/7,4/7,5/7,6/7,8/7; 1/2,2/3,5/6,1,7/6,4/3; 823543*x/46656).
a(n) ~ 7^(7*n+3/2)/(sqrt(Pi)*3^(6*n+5/2)*4^(3*n+1)*n^(3/2)). (End)

A233833 a(n) = 3binomial(7n+3, n)/(7*n+3).

Original entry on oeis.org

1, 3, 24, 253, 3045, 39627, 543004, 7718340, 112752783, 1682460520, 25533901536, 392912889915, 6116090678334, 96133810101609, 1523687678528400, 24324750346691480, 390786855500604195, 6313161418594235271, 102494297789621214400, 1671366110239940499000

Offset: 0

Tim Fulford, Dec 16 2013

nonn

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=7, r=3.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906v1, Discrete Math., 308 (2008), 4660-4669.
Thomas A. Dowling, Catalan Numbers Chapter 7
Clemens Heuberger, Sarah J. Selkirk, and Stephan Wagner, Enumeration of Generalized Dyck Paths Based on the Height of Down-Steps Modulo k, arXiv:2204.14023 [math.CO], 2022.
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.

Cf. A000108, A002296, A233832, A143547, A233834, A130565, A233835, A233907, A233908.

[3*Binomial(7*n+3, n)/(7*n+3): n in [0..30]];

Table[3 Binomial[7 n + 3, n]/(7 n + 3), {n, 0, 30}]

PARI
```
a(n)=3*binomial(7*n+3,n)/(7*n+3);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(7/3))^3+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=7, r=3.
From Ilya Gutkovskiy, Sep 14 2018: (Start)
E.g.f.: 6F6(3/7,4/7,5/7,6/7,8/7,9/7; 2/3,5/6,1,7/6,4/3,3/2; 823543*x/46656).
a(n) ~ 7^(7*n+5/2)/(sqrt(Pi)*3^(6*n+5/2)*4^(3*n+2)*n^(3/2)). (End)

A233907 9binomial(7n+9, n)/(7*n+9).

Original entry on oeis.org

1, 9, 99, 1218, 16065, 222138, 3178140, 46656324, 698868216, 10639125640, 164128169205, 2560224004884, 40314178429707, 639948824981928, 10230035192533800, 164541833894991240, 2660919275605834701, 43239781879996449825, 705687913212419321800, 11561996402992103418000, 190100812111989146008641

Offset: 0

Tim Fulford, Dec 17 2013

nonn

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=7, r=9.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906v1, 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.

Cf. A000108, A002296, A233832, A233833, A143547, A233834, A130565, A233835, A233908.

[9*Binomial(7*n+9, n)/(7*n+9): n in [0..30]];

Table[9 Binomial[7 n + 9, n]/(7 n + 9), {n, 0, 30}]

PARI
```
a(n) = 9*binomial(7*n+9,n)/(7*n+9);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(7/9))^9+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=7, r=9.

D-finite with recurrence 72*n*(6*n+5)*(3*n+2)*(2*n+3)*(3*n+4)*(6*n+7)*a(n) -7*(7*n+4)*(7*n+8)*(7*n+5)*(7*n+2)*(7*n+6)*(7*n+3)*a(n-1)=0. - R. J. Mathar, Nov 22 2024

A233908 10binomial(7n+10,n)/(7*n+10).

Original entry on oeis.org

1, 10, 115, 1450, 19425, 271502, 3915100, 57821940, 870238200, 13298907050, 205811513765, 3218995093860, 50802419972395, 808016193159000, 12938696992921000, 208419656266988904, 3374960506795660365, 54907659530154222000, 897060906625956765000

Offset: 0

Tim Fulford, Dec 17 2013

nonn

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=7, r=10.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906v1, 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. A000108, A002296, A233832 - A233835, A143547, A130565, A233907.

[10*Binomial(7*n+10, n)/(7*n+10): n in [0..30]]; // Vincenzo Librandi, Dec 23 2013

Table[10 Binomial[7 n + 10, n]/(7 n + 10), {n, 0, 40}] (* Vincenzo Librandi, Dec 23 2013 *)

PARI
```
a(n) = 10*binomial(7*n+10,n)/(7*n+10);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(7/10))^10+x*O(x^n)); polcoeff(B, n)}

72*n*(6*n+5)*(3*n+5)*(2*n+3)*(3*n+4)*(6*n+7)*a(n) -7*(7*n+4)*(7*n+8)*(7*n+5)*(7*n+9)*(7*n+6)*(7*n+3)*a(n-1)=0. - R. J. Mathar, Dec 22 2013

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=7, r=10.

A368079 Expansion of (1/x) * Series_Reversion( x * (1-x)^3 * (1-x^2)^3 ).

Original entry on oeis.org

1, 3, 18, 127, 996, 8322, 72644, 654615, 6043455, 56866028, 543368586, 5258196762, 51426990112, 507537393600, 5048033356128, 50549237164615, 509197913456922, 5156339940802941, 52460340305220466, 535976129228082972, 5496745175387480976

Offset: 0

Seiichi Manyama, Jan 18 2024

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..963
Index entries for reversions of series

Cf. A365752, A365856.
Cf. A365878, A365879.
Cf. A130565, A369301.
Cf. A369264, A370271.

my(N=30, x='x+O('x^N)); Vec(serreverse(x*(1-x)^3*(1-x^2)^3)/x)

a(n, s=2, t=3, u=3) = sum(k=0, n\s, binomial(t*(n+1)+k-1, k)*binomial((u+1)*(n+1)-s*k-2, n-s*k))/(n+1);

a(n) = (1/(n+1)) * Sum_{k=0..floor(n/2)} binomial(3*n+k+2,k) * binomial(4*n-2*k+2,n-2*k).
a(n) = (1/(n+1)) * Sum_{k=0..n} (-1)^k * binomial(3*n+k+2,k) * binomial(7*n-k+5,n-k).
a(n) = (1/(n+1)) * [x^n] 1/( (1+x)^3 * (1-x)^6 )^(n+1). - Seiichi Manyama, Feb 16 2024

A386380 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/6)} a(6k) a(n-1-6*k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 7, 15, 24, 34, 45, 57, 70, 154, 253, 368, 500, 650, 819, 1827, 3045, 4495, 6200, 8184, 10472, 23562, 39627, 59052, 82251, 109668, 141778, 320866, 543004, 814506, 1142295, 1533939, 1997688, 4540200, 7718340, 11633440, 16398200, 22137570

Offset: 0

Seiichi Manyama, Jul 20 2025

nonn

Wikipedia, Fuss-Catalan number

Cf. A002296, A233832, A233833, A386392, A233834, A130565.

Cf. A047749, A118968, A124753, A386379, A386396.

A386380 := proc(n)
    option remember ;
    if n = 0 then
        1;
    else
        add(procname(6*k)*procname(n-1-6*k),k=0..floor((n-1)/6)) ;
    end if;
end proc:
seq(A386380(n),n=0..80) ; # R. J. Mathar, Jul 30 2025

apr(n, p, r) = r*binomial(n*p+r, n)/(n*p+r);
a(n) = apr(n\6, 7, n%6+1);

For k=0..5, a(6*n+k) = (k+1) * binomial(7*n+k+1,n)/(7*n+k+1).

G.f. A(x) satisfies A(x) = 1/(1 - x * Product_{k=0..5} A(w^k*x)), where w = exp(Pi*i/3).

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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A233834 a(n) = 5*binomial(7*n+5,n)/(7*n+5).

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A251577 E.g.f.: exp(7*x*G(x)^6) / G(x)^6 where G(x) = 1 + x*G(x)^7 is the g.f. of A002296.

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A233832 a(n) = 2*binomial(7*n+2,n)/(7*n+2).

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A233833 a(n) = 3*binomial(7*n+3, n)/(7*n+3).

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A233907 9*binomial(7*n+9, n)/(7*n+9).

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A233834 a(n) = 5binomial(7n+5,n)/(7*n+5).

A251577 E.g.f.: exp(7xG(x)^6) / G(x)^6 where G(x) = 1 + x*G(x)^7 is the g.f. of A002296.

A233832 a(n) = 2binomial(7n+2,n)/(7*n+2).

A233833 a(n) = 3binomial(7n+3, n)/(7*n+3).

A233907 9binomial(7n+9, n)/(7*n+9).

A233908 10binomial(7n+10,n)/(7*n+10).

A386380 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/6)} a(6k) a(n-1-6*k).