A118971 -id:A118971 - cp's OEIS frontend

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

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)

A329060 4-parking triangle T(r, i, 4) read by rows: T(r, i, k) = (r + 1)^(i-1)binomial(k(r + 1) + r - i - 1, r - i) with k = 4 and 0 <= i <= r.

Original entry on oeis.org

1, 4, 1, 26, 12, 3, 204, 136, 64, 16, 1771, 1540, 1050, 500, 125, 16380, 17550, 15600, 10800, 5184, 1296, 158224, 201376, 220255, 198940, 139258, 67228, 16807, 1577532, 2324784, 3015936, 3351040, 3063808, 2162688, 1048576, 262144, 16112057, 26978328, 40467492, 53298648, 59960979, 55348596, 39326634, 19131876, 4782969

Offset: 0

Stefano Spezia, Nov 03 2019

The k-parking numbers interpolate between the generalized Fuss-Catalan numbers and the number of parking functions (see Yip).

			r/i|    0      1      2      3      4
—————————————————————————————————————
0  |    1
1  |    4      1
2  |   26     12      3
3  |  204    136     64     16
4  | 1771   1540   1050    500    125
...

Stefano Spezia, First 151 rows of the triangle, flattened
Martha Yip, A Fuss-Catalan variation of the caracol flow polytope, arXiv:1910.10060 [math.CO], 2019.

Cf. A000108, A000272, A007318, A118971, A329057, A329058, A329059, A329123 (row sums).

T[r_, i_, k_] := (r + 1)^(i-1)*Binomial[k*(r + 1) + r - i - 1, r - i]; Flatten[Table[T[r,i,4],{r,0,8},{i,0,r}]]

T(r, i, k) = (r + 1)^(i-1)*binomial(k*(r + 1) + r - i - 1, r - i).
T(r, 0, 4) = A118971(r).
T(r, r, 4) = A000272(r + 1).

A371496 G.f. A(x) satisfies A(x) = 1 / (1 - x*A(x) / (1+x))^4.

Original entry on oeis.org

1, 4, 22, 156, 1233, 10420, 92120, 841376, 7876616, 75177492, 728784802, 7156081536, 71024862452, 711383912672, 7181295333306, 72989746391780, 746308443708928, 7671359593228624, 79226966456758424, 821691132077059740, 8554576791134761387

Offset: 0

Seiichi Manyama, Mar 25 2024

nonn

Cf. A001006, A371494, A371495.

Cf. A118971, A349361.

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

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

G.f.: A(x) = B(x)^4 where B(x) is the g.f. of A349361.

A379188 G.f. A(x) satisfies A(x) = 1/((1 - xA(x)^3) (1 - x*A(x))^3).

Original entry on oeis.org

1, 4, 34, 392, 5271, 77530, 1208602, 19620262, 328167191, 5616065633, 97867738285, 1730732539345, 30981439344096, 560293394484145, 10221582080782452, 187884236846039893, 3476266045318846245, 64690833375603622619, 1210026171180264742927, 22736845507710710652858

Offset: 0

Seiichi Manyama, Dec 17 2024

nonn

Cf. A118971, A369617, A379187.
Cf. A379172, A379190, A379191.
Cf. A379186, A379189.

a(n) = sum(k=0, n, binomial(n+3*k+1, k)*binomial(4*n+5*k+2, n-k)/(n+3*k+1));

a(n) = Sum_{k=0..n} binomial(n+3*k+1,k) * binomial(4*n+5*k+2,n-k)/(n+3*k+1).

A386367 a(n) = Sum_{k=0..n-1} binomial(5k,k) binomial(5n-5k-2,n-k-1).

Original entry on oeis.org

0, 1, 13, 163, 2021, 24930, 306655, 3765448, 46182101, 565939603, 6931070490, 84845250370, 1038235255415, 12700966517968, 155336699256808, 1899439862390640, 23222289820948405, 283872591297526505, 3469680960837171415, 42404345427419774621, 518193229118757697930

Offset: 0

Seiichi Manyama, Jul 19 2025

nonn

			(1/5) * log( Sum_{k>=0} binomial(5*k,k)*x^k ) = x + 13*x^2/2 + 163*x^3/3 + 2021*x^4/4 + 4986*x^5 + ...

Cf. A000302, A036829, A308523, A386368.
Cf. A079678, A386566, A386613, A386614.
Cf. A002294, A118971.

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

my(N=30, x='x+O('x^N), g=x*sum(k=0, N, binomial(5*k+3, k)/(k+1)*x^k)); concat(0, Vec(g*(1-g)/(1-5*g)^2))

G.f.: g*(1-g)/(1-5*g)^2 where g*(1-g)^4 = x.
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/5) * log( Sum_{k>=0} binomial(5*k,k)*x^k ).
G.f.: (g-1)/(5-4*g)^2 where g=1+x*g^5.
a(n) = Sum_{k=0..n-1} binomial(5*k-2+l,k) * binomial(5*n-5*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(5*n-1,k).
a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(4*n+k-1,k).

A386566 a(n) = Sum_{k=0..n-1} binomial(5k-1,k) binomial(5n-5k,n-k-1).

Original entry on oeis.org

0, 1, 14, 181, 2284, 28506, 353630, 4370584, 53882392, 663116347, 8150224204, 100073884670, 1227826127020, 15055154471696, 184508186225552, 2260299193652496, 27679951219660080, 338872887728053465, 4147618793911034330, 50753529798492061819, 620942367878256638264

Offset: 0

Seiichi Manyama, Jul 26 2025

nonn

			(1/4) * log( Sum_{k>=0} binomial(5*k-1,k)*x^k ) = x + 7*x^2 + 181*x^3/3 + 571*x^4 + 28506*x^5/5 + ...

Cf. A000346, A062236, A386565, A386567.
Cf. A079678, A386367, A386613, A386614.
Cf. A002294, A118971.

a(n) = sum(k=0, n-1, binomial(5*k-1, k)*binomial(5*n-5*k, n-k-1));

my(N=30, x='x+O('x^N), g=sum(k=0, N, binomial(5*k, k)/(4*k+1)*x^k)); concat(0, Vec(g*(g-1)/(5-4*g)^2))

G.f.: g*(g-1)/(5-4*g)^2 where g=1+x*g^5.
G.f.: g/(1-5*g)^2 where g*(1-g)^4 = x.
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/4) * log( Sum_{k>=0} binomial(5*k-1,k)*x^k ).
a(n) = Sum_{k=0..n-1} binomial(5*k-1+l,k) * binomial(5*n-5*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(5*n,k).
a(n) = Sum_{k=0..n-1} 5^(n-k-1) * binomial(4*n+k,k).
Conjecture D-finite with recurrence 196608*n*(4*n-3)*(2*n-1)*(18270873280*n -32560150837) *(4*n-1)*a(n) +1280*(-1399185802400000*n^5 +1022280893000000*n^4 +17669158913120000*n^3 -48968110172924750*n^2 +49502057719349955*n -17877514345852392)*a(n-1) +125000*(-61298198200000*n^5 +1447969779032500*n^4 -7721498995066250*n^3 +17474948768595875*n^2 -18352567310653770*n +7399184154389181)*a(n-2) +48828125*(5*n-11) *(5*n-14)*(4958243695*n -6717884799) *(5*n-13)*(5*n-12)*a(n-3)=0. - R. J. Mathar, Jul 30 2025

A337292 a(n) = 4binomial(5n,n)/(5*n-1).

Original entry on oeis.org

5, 20, 130, 1020, 8855, 81900, 791120, 7887660, 80560285, 838553320, 8863227100, 94871786100, 1026317094705, 11203116342560, 123243929011680, 1364973221797900, 15207477517956825, 170321264840835900, 1916512328325665070, 21655893753689280120

Offset: 1

Lucas A. Brown, Aug 21 2020

a(n) is the number of lattice paths from (0,0) to (4n,n) using only the steps (1,0) and (0,1) and whose only lattice points on the line y = x/4 are the path's endpoints.

Cf. A002294, A118971, A337291.

Array[4 Binomial[5 #, #]/(5 # - 1) &, 20] (* Michael De Vlieger, Aug 21 2020 *)

a(n) = {4*binomial(5*n,n)/(5*n-1)} \\ Andrew Howroyd, Aug 21 2020

a(n) = 5*A118971(n-1).
G.f.: 5*x*F(x)^4 where F(x) = 1 + x*F(x)^5 is the g.f. of A002294.
D-finite with recurrence 8*n*(4*n-3)*(2*n-1)*(4*n-1)*a(n) -5*(5*n-4)*(5*n-3)*(5*n-2)*(5*n-6)*a(n-1)=0., a(0)=1. - R. J. Mathar, Jan 26 2025
G.f.: -4*4F3(-1/5,1/5,2/5,3/5; 1/4,1/2,3/4; 3125*x/256) . - R. J. Mathar, Aug 10 2025

cp's OEIS Frontend

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

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A329060 4-parking triangle T(r, i, 4) read by rows: T(r, i, k) = (r + 1)^(i-1)*binomial(k*(r + 1) + r - i - 1, r - i) with k = 4 and 0 <= i <= r.

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A371496 G.f. A(x) satisfies A(x) = 1 / (1 - x*A(x) / (1+x))^4.

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A379188 G.f. A(x) satisfies A(x) = 1/((1 - x*A(x)^3) * (1 - x*A(x))^3).

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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).

A329060 4-parking triangle T(r, i, 4) read by rows: T(r, i, k) = (r + 1)^(i-1)binomial(k(r + 1) + r - i - 1, r - i) with k = 4 and 0 <= i <= r.

A379188 G.f. A(x) satisfies A(x) = 1/((1 - xA(x)^3) (1 - x*A(x))^3).

A386367 a(n) = Sum_{k=0..n-1} binomial(5k,k) binomial(5n-5k-2,n-k-1).

A386566 a(n) = Sum_{k=0..n-1} binomial(5k-1,k) binomial(5n-5k,n-k-1).

A337292 a(n) = 4binomial(5n,n)/(5*n-1).