A234573 -id:A234573 - 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

A229963 a(n) = 11binomial(10n + 11, n)/(10*n + 11) .

Original entry on oeis.org

1, 11, 165, 2860, 53900, 1072797, 22188859, 472214600, 10273141395, 227440759700, 5107663394691, 116068178638776, 2664012608972000, 61668340817988135, 1438101958237201950, 33753007927148177360, 796704536753910327114

Offset: 0

Tim Fulford, Oct 04 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(n*p + r,n)/(n*p + r), where p = 10, r = 11.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906 [math.CO], 2007.
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.
Wikipedia, Fuss-Catalan number

Cf. A000108, A059968, A234525, A234526, A234527, A234528, A234529, A234570, A234571, A234573.

Cf. A000245 (k = 3), A006629 (k = 4), A196678 (k = 5), A233668 (k = 6), A233743 (k = 7), A233835 (k = 8), A234467 (k = 9), A232265 (k = 10).

[11*Binomial(10*n+11,n)/(10*n+11) : n in [0..20]]; // Vincenzo Librandi, Jan 10 2014

Table[11/(10 n + 11) Binomial[10 n + 11, n], {n, 0, 40}] (* Vincenzo Librandi, Jan 10 2014 *)

a(n) = 11*binomial(10*n+11,n)/(10*n+11);

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(10/11))^11+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, where p = 10, r = 11.
From _Peter Bala, Oct 16 2015: (Start)
O.g.f. A(x) = 1/x * series reversion (x*C(-x)^11), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. See cross-references for other Fuss-Catalan sequences with o.g.f. 1/x * series reversion (x*C(-x)^k), k = 3 through 11.
A(x)^(1/11) is the o.g.f. for A059968. (End)
D-finite with recurrence: 81*n*(9*n+11)*(9*n+4)*(3*n+2)*(9*n+8)*(9*n+10)*(3*n+1)*(9*n+5)*(9*n+7)*a(n) -800*(10*n+1)*(5*n+1)*(10*n+3)*(5*n+2)*(2*n+1)*(5*n+3)*(10*n+7)*(5*n+4)*(10*n+9)*a(n-1)=0. - R. J. Mathar, Feb 21 2020

Corrected by Vincenzo Librandi, Jan 10 2014

A234571 a(n) = 4binomial(10n+8,n)/(5*n+4).

Original entry on oeis.org

1, 8, 108, 1776, 32430, 632016, 12876864, 270964320, 5843355957, 128462407840, 2868356980060, 64869895026144, 1482877843096650, 34207542810153216, 795318309360948240, 18617396126132233920, 438423206616057162258, 10379232525028947311160, 246878659984195222962220

Offset: 0

Tim Fulford, Dec 28 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), where p = 10, r = 8.

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. A059968, A234525, A234526, A234527, A234528, A234529, A234570, A234573, A059968, A069271, A118970, A212073, A233834, A234465, A234510, A235339.

[4*Binomial(10*n+8, n)/(5*n+4): n in [0..30]];

Table[4 Binomial[10 n + 8, n]/(5 n + 4), {n, 0, 30}]

PARI
```
a(n) = 4*binomial(10*n+8,n)/(5*n+4);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(5/4))^8+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p = 10, r = 8.

O.g.f. A(x) = 1/x * series reversion (x/C(x)^8), where C(x) is the o.g.f. for the Catalan numbers A000108. A(x)^(1/8) is the o.g.f. for A059968. - Peter Bala, Oct 14 2015

A234525 Binomial(10n+2,n)/(5n+1).

Original entry on oeis.org

1, 2, 21, 310, 5330, 99960, 1983049, 40919714, 869304150, 18885977110, 417663940540, 9371084905962, 212791660837756, 4880918206648000, 112925143575796455, 2632162372046272660, 61752662230350642670, 1457074607325333325524

Offset: 0

Tim Fulford, Dec 27 2013

nonn

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), where p=10, 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
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.

Cf. A000108, A059968, A234526, A234527, A234528, A234529, A234570, A234571, A234573, A229963.

[Binomial(10*n+2, n)/(5*n+1): n in [0..30]]; // Vincenzo Librandi, Dec 27 2013

Table[Binomial[10 n + 2, n]/(5 n + 1), {n, 0, 40}] (* Vincenzo Librandi, Dec 27 2013 *)

PARI
```
a(n) = binomial(10*n+2,n)/(5*n+1);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^5)^2+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=10, r=2.

a(n) = 2*binomial(10n+1,n-1)/n for n>0, a(0)=1. [Bruno Berselli, Jan 19 2014]

A251580 E.g.f.: exp(10xG(x)^9) / G(x)^9 where G(x) = 1 + x*G(x)^10 is the g.f. of A059968.

Original entry on oeis.org

1, 1, 10, 280, 13960, 1023760, 99935200, 12226859200, 1801725932800, 310890328768000, 61516405597830400, 13735605457885312000, 3416919943285809280000, 937247149729410729472000, 281051240591439955878400000, 91474949907165746668607488000, 32117399444469103248129863680000

Offset: 0

Paul D. Hanna, Dec 06 2014

nonn

In general, Sum_{k=0..n} m^k * n!/k! * binomial(m*n-k-m, n-k) * (k-1)/(n-1) is for m>1 asymptotic to m^(m*(n-1)-1/2) / (m-1)^((m-1)*(n-1)-1/2) * n^(n-2) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014

			E.g.f.: A(x) = 1 + x + 10*x^2/2! + 280*x^3/3! + 13960*x^4/4! + 1023760*x^5/5! +...
such that A(x) = exp(10*x*G(x)^9) / G(x)^9
where G(x) = 1 + x*G(x)^10 is the g.f. of A059968:
G(x) = 1 + x + 10*x^2 + 145*x^3 + 2470*x^4 + 46060*x^5 + 910252*x^6 +...
Note that
A'(x) = exp(10*x*G(x)^9) = 1 + 10*x + 280*x^2/2! + 13960*x^3/3! +...
LOGARITHMIC DERIVATIVE.
The logarithm of the e.g.f. begins:
log(A(x)) = x + 9*x^2/2 + 252*x^3/3 + 12654*x^4/4 + 933984*x^5/5 +...
and so A'(x)/A(x) = G(x)^9.
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,  10,  280,  13960,  1023760,   99935200,  12226859200, ...];
n=2: [1, 2,  22,  620,  30760,  2243120,  217911520,  26556406400, ...];
n=3: [1, 3,  36, 1026,  50760,  3683880,  356283360,  43256151360, ...];
n=4: [1, 4,  52, 1504,  74344,  5374240,  517647520,  62621962240, ...];
n=5: [1, 5,  70, 2060, 101920,  7344920,  704861200,  84980501600, ...];
n=6: [1, 6,  90, 2700, 133920,  9629280,  921060720, 110691813600, ...];
n=7: [1, 7, 112, 3430, 170800, 12263440, 1169680960, 140152067440, ...];
n=8: [1, 8, 136, 4256, 213040, 15286400, 1454475520, 173796462080, ...]; ...
in which the main diagonal begins (see A251587):
[1, 2, 36, 1504, 101920, 9629280, 1169680960, 173796462080, ...]
and is given by the formula:
[x^n/n!] A(x)^(n+1) = 10^(n-8) * (n+1)^(n-9) * (4782969*n^8 + 50309748*n^7 + 237013938*n^6 + 655232760*n^5 + 1166624361*n^4 + 1374998212*n^3 + 1051760172*n^2 + 479277840*n + 100000000) for n>=0.

Cf. A251590, A251670, A059968, A234573.

Cf. Variants: A243953, A251573, A251574, A251575, A251576, A251577, A251578, A251579.

Flatten[{1,1,Table[Sum[10^k * n!/k! * Binomial[10*n-k-10, 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^10 +x*O(x^n)); n!*polcoeff( exp(10*x*G^9) / G^9, n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = if(n==0, 1, sum(k=0, n, 10^k * n!/k! * binomial(10*n-k-10,n-k)*if(n==1,1/10,(k-1)/(n-1)) ))}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^10 be the g.f. of A059968, then the e.g.f. A(x) of this sequence satisfies:
(1) A'(x)/A(x) = G(x)^9.
(2) A'(x) = exp(10*x*G(x)^9).
(3) A(x) = exp( Integral G(x)^9 dx ).
(4) A(x) = exp( Sum_{n>=1} A234573(n-1)*x^n/n ), where A234573(n-1) = binomial(10*n-2,n)/(9*n-1).
(5) A(x) = F(x/A(x)) where F(x) is the e.g.f. of A251590.
(6) A(x) = Sum_{n>=0} A251590(n)*(x/A(x))^n/n! and
(7) [x^n/n!] A(x)^(n+1) = (n+1)*A251590(n),
where A251590(n) = 10^(n-8) * (n+1)^(n-10) * (4782969*n^8 + 50309748*n^7 + 237013938*n^6 + 655232760*n^5 + 1166624361*n^4 + 1374998212*n^3 + 1051760172*n^2 + 479277840*n + 100000000).
a(n) = Sum_{k=0..n} 10^k * n!/k! * binomial(10*n-k-10, n-k) * (k-1)/(n-1) for n>1.
Recurrence: 81*(3*n-5)*(3*n-4)*(9*n-17)*(9*n-16)*(9*n-14)*(9*n-13)*(9*n-11)*(9*n-10)*(250000*n^8 - 5300000*n^7 + 49332500*n^6 - 263500000*n^5 + 884055975*n^4 - 1909634570*n^3 + 2596659373*n^2 - 2035277286*n + 705468040)*a(n) = 800*(3125000000000*n^17 - 111562500000000*n^16 + 1872125000000000*n^15 - 19618187500000000*n^14 + 143829395937500000*n^13 - 783195370343750000*n^12 + 3281447638218750000*n^11 - 10810863753751875000*n^10 + 28370066880833218750*n^9 - 59681174371832246875*n^8 + 100725400409628775000*n^7 - 135736802338370325750*n^6 + 144424061701272600950*n^5 - 118936947986511839915*n^4 + 73322264536912326596*n^3 - 31942069342168467356*n^2 + 8798129066413437408*n - 1156512281566561920)*a(n-1) + 10000000000*(250000*n^8 - 3300000*n^7 + 19232500*n^6 - 64805000*n^5 + 138543475*n^4 - 193260670*n^3 + 172779013*n^2 - 91243350*n + 22054032)*a(n-2). - Vaclav Kotesovec, Dec 07 2014
a(n) ~ 10^(10*(n-1)-1/2) / 9^(9*(n-1)-1/2) * n^(n-2) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014

A234527 2binomial(10n+4,n)/(5*n+2).

Original entry on oeis.org

1, 4, 46, 704, 12341, 234260, 4685898, 97274544, 2075959314, 45262862788, 1003884090440, 22577660493024, 513698787408521, 11802947663348800, 273471432969603198, 6382396843322710560, 149902629054480517590, 3540479504783000035464

Offset: 0

Tim Fulford, Dec 27 2013

nonn

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), where p=10, r=4.

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, A059968, A234525, A234526, A234528, A234529, A234570, A234571, A234573, A229963.

[2*Binomial(10*n+4, n)/(5*n+2): n in [0..30]]; // Vincenzo Librandi, Dec 28 2013

Table[2 Binomial[10 n + 4, n]/(5 n + 2), {n, 0, 30}] (* Vincenzo Librandi, Dec 28 2013 *)

PARI
```
a(n) = 2*binomial(10*n+4,n)/(5*n+2);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(5/2))^4+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=10, r=4.

A234526 3binomial(10n+3,n)/(10*n+3).

Original entry on oeis.org

1, 3, 33, 496, 8610, 162435, 3235501, 66959532, 1425658806, 31026962395, 687124547340, 15434728080408, 350818684083868, 8053515040969200, 186457795206547635, 4348790005989493960, 102080931442008205230, 2409777235191897422982

Offset: 0

Tim Fulford, Dec 27 2013

nonn

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), where p=10, 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
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.

Cf. A000108, A059968, A234525, A234527, A234528, A234529, A234570, A234571, A234573, A229963.

[3*Binomial(10*n+3, n)/(10*n+3): n in [0..30]]; // Vincenzo Librandi, Dec 28 2013

Table[3 Binomial[10 n + 3, n]/(10 n + 3), {n, 0, 30}] (* Vincenzo Librandi, Dec 28 2013 *)

PARI
```
a(n) = 3*binomial(10*n+3,n)/(10*n+3);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(10/3))^3+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=10, r=3.

A234528 Binomial(10n+5,n)/(2n+1).

Original entry on oeis.org

1, 5, 60, 935, 16555, 316251, 6353760, 132321990, 2830853610, 61841702065, 1373736123760, 30935736733230, 704631080073635, 16204866668942000, 375762274309378440, 8775795659568727020, 206241872189050376550, 4873761343609509542490

Offset: 0

Tim Fulford, Dec 27 2013

nonn

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

Cf. A000108, A059968, A234525, A234526, A234527, A234529, A234570, A234571, A234573, A229963.

[Binomial(10*n+5, n)/(2*n+1): n in [0..30]]; // Vincenzo Librandi, Dec 28 2013

Table[Binomial[10 n + 5, n]/(2 n + 1), {n, 0, 30}] (* Vincenzo Librandi, Dec 28 2013 *)

PARI
```
a(n) = binomial(10*n+5,n)/(2*n+1);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^2)^5+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=10, r=5.

A234529 3binomial(10n+6,n)/(5*n+3).

Original entry on oeis.org

1, 6, 75, 1190, 21285, 409266, 8259888, 172593900, 3701885490, 81033954430, 1803028662435, 40658396849388, 927146157991625, 21342995124948000, 495322997953271580, 11576581508367256920, 272239271289546497046, 6437043284012559773100

Offset: 0

Tim Fulford, Dec 27 2013

nonn

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), where p=10, r=6.

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, A059968, A234525, A234526, A234527, A234528, A234570, A234571, A234573, A229963.

[3*Binomial(10*n+6, n)/(5*n+3): n in [0..30]]; // Vincenzo Librandi, Dec 27 2013

Table[3 Binomial[10 n + 6, n]/(5 n + 3), {n, 0, 30}] (* Vincenzo Librandi, Dec 27 2013 *)

PARI
```
a(n) = 3*binomial(10*n+6,n)/(5*n+3);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(5/3))^6+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=10, r=6.

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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A229963 a(n) = 11*binomial(10*n + 11, n)/(10*n + 11) .

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A234571 a(n) = 4*binomial(10*n+8,n)/(5*n+4).

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A234525 Binomial(10*n+2,n)/(5*n+1).

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A251580 E.g.f.: exp(10*x*G(x)^9) / G(x)^9 where G(x) = 1 + x*G(x)^10 is the g.f. of A059968.

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A234527 2*binomial(10*n+4,n)/(5*n+2).

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A229963 a(n) = 11binomial(10n + 11, n)/(10*n + 11) .

A234571 a(n) = 4binomial(10n+8,n)/(5*n+4).

A234525 Binomial(10n+2,n)/(5n+1).

A251580 E.g.f.: exp(10xG(x)^9) / G(x)^9 where G(x) = 1 + x*G(x)^10 is the g.f. of A059968.

A234527 2binomial(10n+4,n)/(5*n+2).

A234526 3binomial(10n+3,n)/(10*n+3).

A234528 Binomial(10n+5,n)/(2n+1).

A234529 3binomial(10n+6,n)/(5*n+3).