A118969 -id:A118969 - cp's OEIS frontend

A118970 a(n) = 3*binomial(5n+2,n)/(4n+3).

1, 3, 18, 136, 1155, 10530, 100688, 996336, 10116873, 104819165, 1103722620, 11777187240, 127067830773, 1383914371728, 15194457001440, 167996704221280, 1868870731122405, 20903064321375315, 234927317665726686

Offset: 0

Paul Barry, May 07 2006

A quadrisection of A118968.

Convolved with A118969 (1, 2, 11, 80, 665, ...) = A002294: (1, 5, 35, 285, 2530, ...) - Gary W. Adamson, Nov 07 2011

R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section B33.

Michael De Vlieger, Table of n, a(n) for n = 0..924
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.
Henri Muehle and Philippe Nadeau, A Poset Structure on the Alternating Group Generated by 3-Cycles, arXiv:1803.00540 [math.CO], 2018.

Cf. A118969, A002294, A000108, A163456.

ogf := series(RootOf(A = 1 + x * A^5,A)^3, x=0, 30); # Mark van Hoeij, Apr 22 2013

Array[3 Binomial[5 # + 2, #]/(4 # + 3) &, 19, 0] (* Michael De Vlieger, May 30 2018 *)
CoefficientList[Series[HypergeometricPFQ[{3/5,4/5,6/5,7/5},{1,5/4,3/2,7/4},(5^5/4^4)x],{x,0,18}],x]Range[0,18]! (* Stefano Spezia, Oct 01 2024 *)

a(n)=3*binomial(5*n+2,n)/(4*n+3); \\ Joerg Arndt, Apr 23 2013

G.f.: F^3 where F is the g.f. of A002294. - Mark van Hoeij, Apr 23 2013
8*n*(4*n+1)*(2*n+1)*(4*n+3)*a(n) -5*(5*n+1)*(5*n+2)*(5*n-2)*(5*n-1)*a(n-1)=0. - R. J. Mathar, Dec 02 2014
From Peter Bala, Oct 08 2015: (Start)
O.g.f. A(x) = (1/x) * series reversion ( x/C(x)^3 ), where C(x) = (1 - sqrt(1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108.
(1/3)*x*A'(x)/A(x) = x + 9*x^2 + 91*x^3 + 969*x^4 + ... is the o.g.f. for A163456. (End)
E.g.f.: hypergeom([3/5, 4/5, 6/5, 7/5], [1, 5/4, 3/2, 7/4], (5^5/4^4)*x). - Stefano Spezia, Oct 01 2024

A233668 a(n) = 6binomial(5n + 6,n)/(5*n + 6).

Original entry on oeis.org

1, 6, 45, 380, 3450, 32886, 324632, 3290040, 34034715, 357919100, 3815041230, 41124015036, 447534498320, 4910258796240, 54257308779600, 603260892430960, 6744185681876505, 75764901779438850, 854867886710698755, 9683529727259434200

Offset: 0

Tim Fulford, Dec 14 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(n*p + r, n)/(n*p + r); this is the case p = 5, r = 6.

C. H. Pah, M. R. Wahiddin, Combinatorial Interpretation of Raney Numbers and Tree Enumerations, Open Journal of Discrete Mathematics, 2015, 5, 1-9; http://www.scirp.org/journal/ojdm; http://dx.doi.org/10.4236/ojdm.2015.51001

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, A002294, A118969, A143546, A118971, A233669, A233736, A233737, A233738.

Cf. A000245 (k = 3), A006629 (k = 4), A196678 (k = 5), A233743 (k = 7), A233835 (k = 8), A234467 (k = 9), A232265 (k = 10), A229963 (k = 11).

[6*Binomial(5*n+6,n)/(5*n+6): n in [0..30]];

Table[6 Binomial[5 n + 6, n]/(5 n + 6), {n, 0, 30}]

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

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

G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, here p = 5, r = 6.
From _Peter Bala, Oct 16 2015: (Start)
O.g.f. A(x) = 1/x * series reversion (x*C(-x)^6), 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)^(1/6) is the o.g.f. for A002294. (End)
D-finite with recurrence 8*n*(4*n+5)*(2*n+3)*(4*n+3)*a(n) -5*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*a(n-1)=0. - R. J. Mathar, Nov 22 2024

A118968 a(4n+k) = (k+1)*binomial(5n+k,n)/(4n+k+1), k=0..3.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 3, 4, 5, 11, 18, 26, 35, 80, 136, 204, 285, 665, 1155, 1771, 2530, 5980, 10530, 16380, 23751, 56637, 100688, 158224, 231880, 556512, 996336, 1577532, 2330445, 5620485, 10116873, 16112057, 23950355, 57985070, 104819165, 167710664, 250543370, 608462470

Offset: 0

Paul Barry, May 07 2006

Row sums of Riordan array (1,x(1-x^4))^(-1).

F. Hering et al., The enumeration of stack polytopes and simplicial clusters, Discrete Math., 40 (1982), 203-217.

Cf. A124753, A002294, A118969, A118970, A118971.

Table[k=Mod[n,4];(k+1)Binomial[(5n-k)/4,(n-k)/4]/(n+1),{n,0,40}] (* Robert A. Russell, Mar 14 2024 *)

{a(n)=local(A=1+x+x*O(x^n));for(i=1,n,A=1+x*A^2*subst(A,x,-x)*subst(A,x,I*x)*subst(A,x,-I*x));polcoeff(A,n)} \\ Paul D. Hanna, Jun 04 2012

{a(n)=local(A=1+x);for(i=1,n,A=1+x*A*exp(sum(m=1,n\4,4*polcoeff(log(A+x*O(x^n)),4*m)*x^(4*m))+x*O(x^n)));polcoeff(A,n)} \\ Paul D. Hanna, Jun 04 2012

apr(n, p, r) = r*binomial(n*p+r, n)/(n*p+r);
a(n) = apr(n\4, 5, n%4+1); \\ Seiichi Manyama, Jul 20 2025

a(4n) = A002294(n), a(4n+1) = A118969(n), a(4n+2) = A118970(n), a(4n+3) = A118971(n).
G.f. satisfies: A(x) = 1 + x*A(x)^2*A(-x)*A(I*x)*A(-I*x). - Paul D. Hanna, Jun 04 2012
G.f. satisfies: A(x) = 1 + x*A(x)*G(x^4) where G(x) = 1 + x*G(x)^5 is the g.f. of A002294. - Paul D. Hanna, Jun 04 2012
From Robert A. Russell, Mar 14 2024: (Start)
G.f.: G(z^4) + z*G(z^4)^2 + z^2*G(z^4)^3 + z^3*G(z^4)^4, where G(z) = 1 + z*G(z)^5 is the g.f. for A002294.
G.f.: E(1)(t*E(5)(t^4)) (fifth entry in Table 3), where E(d)(t) is defined in formula 3 of Hering link. (End)
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/4)} a(4*k) * a(n-1-4*k). - Seiichi Manyama, Jul 07 2025

A233738 2binomial(5n+10, n)/(n+2).

Original entry on oeis.org

1, 10, 95, 920, 9135, 92752, 959595, 10084360, 107375730, 1156073100, 12565671261, 137702922560, 1519842008360, 16880051620320, 188519028884675, 2115822959020080, 23851913523156675, 269958280013904870, 3066451080298820830, 34946186787944832400

Offset: 0

Tim Fulford, Dec 15 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=5, r=10.

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.

Cf. A000108, A002294, A004344, A118969, A118971, A143546, A233668, A233669, A233736, A233737.

[2*Binomial(5*n+10, n)/(n+2): n in [0..30]]; // Vincenzo Librandi, Dec 16 2013

A233738:=n->2*binomial(5*n+10,n)/(n+2): seq(A233738(n), n=0..30); # Wesley Ivan Hurt, Sep 07 2014

Table[2 Binomial[5 n + 10, n]/(n + 2), {n, 0, 40}] (* Vincenzo Librandi, Dec 16 2013 *)

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

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(1/2))^10+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=5, r=10.
a(n) = 2*A004344(n)/(n+2). - Wesley Ivan Hurt, Sep 07 2014
G.f.: hypergeom([2, 11/5, 12/5, 13/5, 14/5], [11/4, 3, 13/4, 7/2], (3125/256)*x). - Robert Israel, Sep 07 2014
D-finite with recurrence 8*(2*n+5)*(4*n+7)*(n+2)*(4*n+9)*a(n) -(n+1)*(13877*n^3+45630*n^2+46579*n+14034)*a(n-1) +210*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*a(n-2)=0. - R. J. Mathar, Nov 22 2024
D-finite with recurrence 8*n*(2*n+5)*(4*n+7)*(n+2)*(4*n+9)*a(n) -5*(5*n+6)*(5*n+7)*(5*n+8)*(5*n+9)*(n+1)*a(n-1)=0. - R. J. Mathar, Nov 22 2024

A233669 a(n) = 7binomial(5n+7, n)/(5*n+7).

Original entry on oeis.org

1, 7, 56, 490, 4550, 44051, 439824, 4496388, 46834095, 495260150, 5303177880, 57385471962, 626548297648, 6893781417320, 76362138282400, 850867975145160, 9530515916642385, 107249427630005661, 1211964598880990640, 13747501038498835300

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=5, r=7.

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, A002294, A118969, A143546, A118971, A233668, A233736, A233737, A233738.

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

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

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

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

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, here p=5, r=7.
From Ilya Gutkovskiy, Sep 14 2018: (Start)
E.g.f.: 4F4(7/5,8/5,9/5,11/5; 1,9/4,5/2,11/4; 3125*x/256).
a(n) ~ 7*5^(5*n+13/2)/(sqrt(Pi)*2^(8*n+31/2)*n^(3/2)). (End)

A233736 a(n) = 8binomial(5n + 8, n)/(5*n + 8).

Original entry on oeis.org

1, 8, 68, 616, 5850, 57536, 581196, 5995184, 62891499, 668922800, 7197169980, 78195588168, 856708896784, 9454328800896, 104997940138300, 1172624772468960, 13161188646791865, 148375147999406328, 1679436658449372744, 19078164706488179600

Offset: 0

Tim Fulford, Dec 15 2013

nonn

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

Cf. A000108, A002294, A118969, A143546, A118971, A233668, A233669, A233737, A233738.

[8*Binomial(5*n+8,n)/(5*n+8): n in [0..30]]; // Vincenzo Librandi, Dec 16 2013

Table[8 Binomial[5 n + 8, n]/(5 n + 8), {n, 0, 40}] (* Vincenzo Librandi, Dec 16 2013 *)

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

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

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=5, r=8.
From Ilya Gutkovskiy, Sep 14 2018: (Start)
E.g.f.: 5F5(8/5,9/5,2,11/5,12/5; 1,9/4,5/2,11/4,3; 3125*x/256).
a(n) ~ 5^(5*n+15/2)/(sqrt(Pi)*2^(8*n+29/2)*n^(3/2)). (End)

A233737 a(n) = 9binomial(5n+9, n)/(5*n+9).

Original entry on oeis.org

1, 9, 81, 759, 7371, 73656, 752913, 7838298, 82832706, 886322710, 9583986555, 104568156819, 1149793519368, 12728471356944, 141747219186705, 1586867219265060, 17848735288114995, 201607141031660871, 2285899896222757346, 26008027474874327190, 296840444852078282610, 3397721117411729991960

Offset: 0

Tim Fulford, Dec 15 2013

nonn

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=5, 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, A002294, A118969, A143546, A118971, A233668, A233669, A233736, A233738.

[9*Binomial(5*n+9,n)/(5*n+9): n in [0..30]];

Table[9 Binomial[5 n + 9, n]/(5 n + 9), {n, 0, 30}]

PARI
```
a(n) = 9*binomial(5*n+9,n)/(5*n+9);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(5/9))^9+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, here p=5, r=9.
From Ilya Gutkovskiy, Sep 14 2018: (Start)
E.g.f.: 5F5(9/5,2,11/5,12/5,13/5; 1,5/2,11/4,3,13/4; 3125*x/256).
a(n) ~ 9*5^(5*n+17/2)/(sqrt(Pi)*2^(8*n+39/2)*n^(3/2)). (End)

A379284 G.f. A(x) satisfies A(x) = 1/((1 - xA(x)^2) (1 - x*A(x)^4)).

Original entry on oeis.org

1, 2, 15, 158, 1943, 26099, 371128, 5491868, 83692617, 1304579981, 20703125143, 333366138381, 5433036837372, 89448269251685, 1485469625972490, 24854484773368344, 418581393456669989, 7090045259711970090, 120706208890692261466, 2064356606197948427512, 35449776962011108029539

Offset: 0

Seiichi Manyama, Dec 19 2024

nonn

Cf. A118969, A199475, A379209, A379285, A379286, A379287.

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

a(n) = Sum_{k=0..n} binomial(2*n+3*k+1,k) * binomial(3*n+k,n-k)/(2*n+3*k+1).

A379287 G.f. A(x) satisfies A(x) = 1/((1 - xA(x)^2) (1 - x*A(x)^6)).

Original entry on oeis.org

1, 2, 19, 268, 4477, 82110, 1597963, 32402460, 677152153, 14481799261, 315417278757, 6972246638416, 156017257712825, 3527275634678216, 80447862652931941, 1848737311902300600, 42766087499793329349, 995043161703028219128, 23271045049097437148389

Offset: 0

Seiichi Manyama, Dec 19 2024

nonn

Cf. A118969, A199475, A379209, A379284, A379285, A379286.

a(n) = sum(k=0, n, binomial(2*n+5*k+1, k)*binomial(3*n+3*k, n-k)/(2*n+5*k+1));

a(n) = Sum_{k=0..n} binomial(2*n+5*k+1,k) * binomial(3*n+3*k,n-k)/(2*n+5*k+1).

A379285 G.f. A(x) satisfies A(x) = 1/((1 - xA(x)^2) (1 - x*A(x)^3)).

Original entry on oeis.org

1, 2, 13, 115, 1175, 13052, 153115, 1866599, 23414063, 300238945, 3917984904, 51862207151, 694670871393, 9398137507922, 128235826442635, 1762706644013297, 24386388751113511, 339295523459625535, 4744546261930628062, 66644485202547680010, 939916204595095866644

Offset: 0

Seiichi Manyama, Dec 19 2024

nonn

Cf. A118969, A199475, A379209, A379284, A379286, A379287.

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

a(n) = Sum_{k=0..n} binomial(2*n+2*k+1,k) * binomial(3*n,n-k)/(2*n+2*k+1).

cp's OEIS Frontend

A118970 a(n) = 3*binomial(5n+2,n)/(4n+3).

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A233668 a(n) = 6*binomial(5*n + 6,n)/(5*n + 6).

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A118968 a(4n+k) = (k+1)*binomial(5n+k,n)/(4n+k+1), k=0..3.

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A233738 2*binomial(5*n+10, n)/(n+2).

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A233669 a(n) = 7*binomial(5*n+7, n)/(5*n+7).

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A233736 a(n) = 8*binomial(5*n + 8, n)/(5*n + 8).

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A233668 a(n) = 6binomial(5n + 6,n)/(5*n + 6).

A233738 2binomial(5n+10, n)/(n+2).

A233669 a(n) = 7binomial(5n+7, n)/(5*n+7).

A233736 a(n) = 8binomial(5n + 8, n)/(5*n + 8).

A233737 a(n) = 9binomial(5n+9, n)/(5*n+9).

A379284 G.f. A(x) satisfies A(x) = 1/((1 - xA(x)^2) (1 - x*A(x)^4)).

A379287 G.f. A(x) satisfies A(x) = 1/((1 - xA(x)^2) (1 - x*A(x)^6)).

A379285 G.f. A(x) satisfies A(x) = 1/((1 - xA(x)^2) (1 - x*A(x)^3)).