A212072 -id:A212072 - cp's OEIS frontend

A212073 G.f. satisfies: A(x) = (1 + x*A(x)^(3/2))^4.

1, 4, 30, 280, 2925, 32736, 383838, 4654320, 57887550, 734405100, 9467075926, 123648163392, 1632743088275, 21761329287600, 292362576381900, 3955219615609056, 53834425161872586, 736687428853685400, 10129401435828605700, 139876690363085200200

Offset: 0

Paul D. Hanna, Apr 29 2012

Fuss-Catalan sequence is a(n,p,r) = r*binomial(p*n + r, n)/(p*n + r); this is the case p = 6, r = 4. The o.g.f. B(x) of the Fuss_catalan sequence a(n,p,r) satisfies B(x) = {1 + x*B(x)^(p/r)}^r. - Peter Bala, Oct 14 2015

			G.f.: A(x) = 1 + 4*x + 30*x^2 + 280*x^3 + 2925*x^4 + 32736*x^5 +...
Related expansions:
A(x)^(3/2) = 1 + 6*x + 51*x^2 + 506*x^3 + 5481*x^4 +...+ A002295(n+1)*x^n +...
A(x)^(1/4) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 +...+ A002295(n)*x^n +...

Wikipedia, Fuss-Catalan number

Cf. A002295, A212071, A212072, A130564, A069271, A118970, A233834, A234465, A234510, A234571, A235339.

m = 20; A[_] = 0;
Do[A[x_] = (1 + x*A[x]^(3/2))^4 + O[x]^m, {m}];
CoefficientList[A[x], x] (* Jean-François Alcover, Oct 20 2019 *)

{a(n)=binomial(6*n+4,n) * 4/(6*n+4)}
for(n=0, 40, print1(a(n), ", "))

{a(n)=local(A=1+4*x); for(i=1, n, A=(1+x*A^(3/2))^4+x*O(x^n)); polcoeff(A, n)}

a(n) = 4*binomial(6*n+4,n)/(6*n+4).
G.f. A(x) = G(x)^4 where G(x) = 1 + x*G(x)^6 is the g.f. of A002295.
O.g.f. A(x) = 1/x * series reversion (x/C(x)^4), where C(x) is the o.g.f. for the Catalan numbers A000108. - Peter Bala, Oct 14 2015
D-finite with recurrence 5*n*(5*n+1)*(5*n+2)*(5*n+3)*(5*n+4)*a(n) -72*(6*n-1)*(3*n-1)*(2*n+1)*(3*n+1)*(6*n+1)*a(n-1)=0. - R. J. Mathar, Nov 22 2024

A212071 G.f. satisfies: A(x) = (1 + x*A(x)^3)^2.

Original entry on oeis.org

1, 2, 13, 114, 1150, 12586, 145299, 1741844, 21475146, 270570300, 3468352701, 45089941936, 593082894768, 7878407177270, 105542811922950, 1424267372100456, 19343105144742098, 264182048662182420, 3626176386241346070, 49995713597946235350, 692084935397470961346

Offset: 0

Paul D. Hanna, Apr 29 2012

The two parameter Fuss-Catalan sequence is A(n,p,r) := r*binomial(n*p + r, n)/(n*p + r), with o.g.f. G(p,r,x) = G(x) satisfying G(x) = {1 + x*G(x)^(p/r)}^r ; this is the case p = 6, r = 2. - Peter Bala, Oct 14 2015

			G.f.: A(x) = 1 + 2*x + 13*x^2 + 114*x^3 + 1150*x^4 + 12586*x^5 +...
Related expansions:
A(x)^3 = 1 + 6*x + 51*x^2 + 506*x^3 + 5481*x^4 +...+ A002295(n+1)*x^n +...
A(x)^(1/2) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 +...+ A002295(n)*x^n +...

Michael De Vlieger, Table of n, a(n) for n = 0..856
Gi-Sang Cheon, S.-T. Jin, and L. W. Shapiro, A combinatorial equivalence relation for formal power series, Linear Algebra and its Applications, Volume 491, 15 February 2016, Pages 123-137.
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.
Wikipedia, Fuss-Catalan number

Cf. A002295, A212072, A212073, A130564.

Table[c=6n+2;(2*Binomial[c,n])/c,{n,0,20}] (* Harvey P. Dale, Oct 14 2013 *)

{a(n)=binomial(6*n+2,n) * 2/(6*n+2)}
for(n=0, 40, print1(a(n), ", "))

{a(n)=local(A=1+2*x); for(i=1, n, A=(1+x*A^3)^2+x*O(x^n)); polcoeff(A, n)}

a(n) = 2*binomial(6*n+2,n)/(6*n+2).
G.f.: A(x) = G(x)^2 where G(x) = 1 + x*G(x)^6 is the g.f. of A002295.
a(n) = 2*binomial(6n+1, n-1)/n for n>0, a(0)=1. [Bruno Berselli, Jan 19 2014]
A(x^2) = 1/x * series reversion (x/C(x^2)^2), where C(x) = (1 - sqrt( 1 - 4*x))/(2*x) is the o.g.f. for the Catalan numbers A000108. - Peter Bala, Oct 14 2015
D-finite with recurrence 5*n*(5*n+1)*(5*n+2)*(5*n-2)*(5*n-1)*a(n) -72*(6*n-1)*(3*n-1)*(2*n-1)*(3*n-2)*(6*n+1)*a(n-1)=0. - R. J. Mathar, Nov 22 2024

A233743 a(n) = 7binomial(6n + 7, n)/(6*n + 7).

Original entry on oeis.org

1, 7, 63, 644, 7105, 82467, 992446, 12271512, 154962990, 1990038435, 25909892008, 341225775072, 4537563627415, 60842326873230, 821692714673340, 11167153485624304, 152610018401940330, 2095863415900961490, 28910564819681953485, 400379714692751795820

Offset: 0

Tim Fulford, Dec 15 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(n*p + r, n)/(n*p + r); this is the case p = 6, r = 7.

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, A002295, A212071, A212072, A212073, A130564, A233827, A233829, A233830.

Cf. A000245 (k = 3), A006629 (k = 4), A196678 (k = 5), A233668 (k = 6), A233835 (k = 8), A234467 (k = 9), A232265 (k = 10), A229963 (k = 11).

[7*Binomial(6*n+7, n)/(6*n+7): n in [0..30]]; // Vincenzo Librandi, Dec 16 2013

Table[7 Binomial[6 n + 7, n]/(6 n + 7), {n, 0, 40}] (* Vincenzo Librandi, Dec 16 2013 *)

PARI
```
a(n) = 7*binomial(6*n+7,n)/(6*n+7);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(6/7))^7+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, where p = 6, r = 7.
From _Peter Bala, Oct 16 2015: (Start)
O.g.f. A(x) = 1/x * series reversion (x*C(-x)^7), 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)^(1/7) is the o.g.f. for A002295. (End)

More terms from Vincenzo Librandi, Dec 16 2013

A233827 a(n) = 8binomial(6n+8,n)/(6*n+8).

Original entry on oeis.org

1, 8, 76, 800, 8990, 105672, 1283464, 15981504, 202927725, 2617624680, 34206162848, 451872681728, 6024664312030, 80964348872400, 1095590286231120, 14915165412813184, 204140673966231870, 2807362363541687280, 38772186055550141700

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=6, 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, A002295, A212071, A212072, A212073, A130564, A233743, A233829, A233830.

[8*Binomial(6*n+8, n)/(6*n+8): n in [0..30]];

Table[8 Binomial[6 n + 8, n]/(6 n + 8), {n, 0, 30}]

PARI
```
a(n) = 8*binomial(6*n+8,n)/(6*n+8);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(6/8))^8+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=6, r=8.
From Ilya Gutkovskiy, Sep 14 2018: (Start)
E.g.f.: 5F5(4/3,3/2,5/3,11/6,13/6; 1,9/5,11/5,12/5,13/5; 46656*x/3125).
a(n) ~ 3^(6*n+15/2)*4^(3*n+5)/(sqrt(Pi)*5^(5*n+17/2)*n^(3/2)). (End)
D-finite with recurrence 5*n*(5*n+6)*(5*n+7)*(5*n+8)*(5*n+4)*a(n) -72*(6*n+5)*(3*n+2)*(2*n+1)*(3*n+1)*(6*n+7)*a(n-1)=0. - R. J. Mathar, Nov 22 2024

A233829 a(n) = 3binomial(6n+9,n)/(2*n+3).

Original entry on oeis.org

1, 9, 90, 975, 11160, 132867, 1629012, 20430900, 260907075, 3381098545, 44352058608, 587787511779, 7858257798300, 105855415586550, 1435361957277480, 19576154604317304, 268364706225271110, 3695862686045572350, 51108790709588823150

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=6, 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, A002295, A212071, A212072, A212073, A130564, A233743, A233827, A233830.

[3*Binomial(6*n+9, n)/(2*n+3): n in [0..30]];

Table[3 Binomial[6 n + 9, n]/(2 n + 3), {n, 0, 30}]

PARI
```
a(n) = 3*binomial(6*n+9,n)/(2*n+3);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(2/3))^9+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=6, r=9.
From Ilya Gutkovskiy, Sep 14 2018: (Start)
E.g.f.: 5F5(3/2,5/3,11/6,13/6,7/3; 1,11/5,12/5,13/5,14/5; 46656*x/3125).
a(n) ~ 3^(6*n+21/2)*4^(3*n+4)/(sqrt(Pi)*5^(5*n+19/2)*n^(3/2)). (End)

A233830 a(n) = 5binomial(6n+10,n)/(3*n+5).

Original entry on oeis.org

1, 10, 105, 1170, 13640, 164502, 2036265, 25727800, 330482295, 4303216330, 56672074888, 753573733050, 10103474312100, 136435868978220, 1854009194816745, 25333847134998864, 347880174736462550, 4798137522234602700, 66441427922465470095, 923346006310186106010, 12873823246049001482400

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=6, 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.

Cf. A000108, A002295, A212071, A212072, A212073, A130564, A233743, A233827, A233829.

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

Table[5 Binomial[6 n + 10, n]/(3 n + 5), {n, 0, 30}]

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

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(3/5))^10+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, here p=6, r=10.
From Ilya Gutkovskiy, Sep 14 2018: (Start)
E.g.f.: 6F6(5/3,11/6,2,13/6,7/3,5/2; 1,11/5,12/5,13/5,14/5,3; 46656*x/3125).
a(n) ~ 3^(6*n+19/2)*4^(3*n+5)/(sqrt(Pi)*5^(5*n+19/2)*n^(3/2)). (End)

A386379 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/5)} a(5k) a(n-1-5*k).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 3, 4, 5, 6, 13, 21, 30, 40, 51, 114, 190, 280, 385, 506, 1150, 1950, 2925, 4095, 5481, 12586, 21576, 32736, 46376, 62832, 145299, 250971, 383838, 548340, 749398, 1741844, 3025308, 4654320, 6690585, 9203634, 21475146, 37456650, 57887550

Offset: 0

Seiichi Manyama, Jul 20 2025

nonn

Wikipedia, Fuss-Catalan number

Cf. A002295, A212071, A212072, A212073, A130564.

Cf. A047749, A118968, A124753, A386380.

apr(n, p, r) = r*binomial(n*p+r, n)/(n*p+r);
a(n) = apr(n\5, 6, n%5+1);

For k=0..4, a(5*n+k) = (k+1) * binomial(6*n+k+1,n)/(6*n+k+1).

G.f. A(x) satisfies A(x) = 1/(1 - x * Product_{k=0..4} A(w^k*x)), where w = exp(2*Pi*i/5).

cp's OEIS Frontend

A212073 G.f. satisfies: A(x) = (1 + x*A(x)^(3/2))^4.

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A212071 G.f. satisfies: A(x) = (1 + x*A(x)^3)^2.

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A233743 a(n) = 7*binomial(6*n + 7, n)/(6*n + 7).

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A233827 a(n) = 8*binomial(6*n+8,n)/(6*n+8).

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A233829 a(n) = 3*binomial(6*n+9,n)/(2*n+3).

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PARI

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A233830 a(n) = 5*binomial(6*n+10,n)/(3*n+5).

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Magma

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PARI

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A233743 a(n) = 7binomial(6n + 7, n)/(6*n + 7).

A233827 a(n) = 8binomial(6n+8,n)/(6*n+8).

A233829 a(n) = 3binomial(6n+9,n)/(2*n+3).

A233830 a(n) = 5binomial(6n+10,n)/(3*n+5).

A386379 a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/5)} a(5k) a(n-1-5*k).