A006632 -id:A006632 - cp's OEIS frontend

A124753 a(3n+k) = (k+1)*binomial(4n+k, n)/(3n+k+1), where k is n reduced mod 3.

1, 1, 1, 1, 2, 3, 4, 9, 15, 22, 52, 91, 140, 340, 612, 969, 2394, 4389, 7084, 17710, 32890, 53820, 135720, 254475, 420732, 1068012, 2017356, 3362260, 8579560, 16301164, 27343888, 70068713, 133767543, 225568798, 580034052, 1111731933, 1882933364, 4855986044, 9338434700

Offset: 0

Paul Barry, Nov 06 2006

Row sums of Riordan array (1,x(1-x^3))^(-1). Also row sums of A124752.
a(n) is the number of ordered trees (A000108) with n vertices in which every non-leaf non-root vertex has exactly two children that are leaves. For example, a(4) counts the 2 trees
  \ /
   |   and  \|/ . - David Callan, Aug 22 2014

J.-B. Priez and A. Virmaux, Non-commutative Frobenius characteristic of generalized parking functions: Application to enumeration, arXiv:1411.4161 [math.CO], 2014-2015.

Cf. A084080, A002293, A069271 (trisection), A006632 (trisection).

Cf. A047749, A118968.

A124753 := proc(n)
    local k,np;
    k := modp(n,3) ;
    np := floor(n/3) ;
    (k+1)*binomial(np+n,np)/(n+1) ;
end proc:
seq(A124753(n),n=0..40) ; # R. J. Mathar, Oct 30 2014

a[n_] := Module[{q, k}, {q, k} = QuotientRemainder[n, 3]; (k+1)*Binomial[4q + k, q]/(3q + k + 1)];
Table[a[n], {n, 0, 34}] (* Jean-François Alcover, Nov 20 2017 *)

{a(n)=local(A=1+x); for(i=1,n,A=1+x*A*exp(sum(m=1,n\3,3*polcoeff(log(A+x*O(x^n)),3*m)*x^(3*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\3, 4, n%3+1); \\ Seiichi Manyama, Jul 20 2025

a(3n) = A002293(n), a(3n+1) = A069271(n), a(3n+2) = A006632(n+1).
a(n) = ((mod(n,3)+1)*C(4*floor(n/3)+mod(n,3), floor(n/3))/ (3*floor(n/3) + 1 + mod(n, 3))). - Paul Barry, Dec 14 2006
G.f. satisfies: A(x) = 1 + x*A(x)^2*A(w*x)*A(w^2*x), where w = exp(2*Pi*I/3). - Paul D. Hanna, Jun 04 2012
G.f. satisfies: A(x) = 1 + x*A(x)*G(x^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293. - Paul D. Hanna, Jun 04 2012
Conjecture: +8019*n*(n-1)*(n+1)*a(n) +17496*n*(n-1)*(n-3)*a(n-1) +2592*(3*n-5)*(n-1)*(3*n-16)*a(n-2) +216*(-224*n^3+48*n^2+3926*n-6331)*a(n-3) +576*(-288*n^3+2448*n^2-6558*n+5443)*a(n-4) +768*(-288*n^3+3600*n^2-14878*n+20375)*a(n-5) -8192*(4*n-23)*(2*n-11)*(4*n-21)*a(n-6)=0. - R. J. Mathar, Oct 30 2014
a(0) = 1; a(n) = Sum_{k=0..floor((n-1)/3)} a(3*k) * a(n-1-3*k). - Seiichi Manyama, Jul 07 2025

A006634 a(n) = 3binomial(4n+8, n)/(n+3).

Original entry on oeis.org

1, 9, 72, 570, 4554, 36855, 302064, 2504304, 20974005, 177232627, 1509395976, 12943656180, 111676661460, 968786892675, 8445123522144, 73940567860896, 649942898236596, 5733561315124260, 50744886833898400, 450461491952952690, 4009721145437152530, 35782256673785401065

Offset: 0

Simon Plouffe

Former name: From generalized Catalan numbers.

H. M. Finucan, Some decompositions of generalized Catalan numbers, pp. 275-293 of Combinatorial Mathematics IX. Proc. Ninth Australian Conference (Brisbane, August 1981). Ed. E. J. Billington, S. Oates-Williams and A. P. Street. Lecture Notes Math., 952. Springer-Verlag, 1982.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Simon Plouffe, Approximations of generating functions and a few conjectures, arXiv:0911.4975 [math.NT], 2002.

Cf. A006630, A006631, A006632, A006633, A006635, A006636, A006637.

A006634:= func< n | 3*Binomial(4*n+8,n)/(n+3) >;
[A006634(n): n in [0..40]]; // G. C. Greubel, Sep 01 2025

series(RootOf(g = 1+x*g^4,g)^9, x=0, 30); # Mark van Hoeij, Apr 22 2013

f[x_] := HypergeometricPFQ[ {9/4, 5/2, 11/4, 3}, {10/3, 11/3, 4}, 256/27*x]; Series[f[x], {x, 0, 16}] // CoefficientList[#, x]& (* Jean-François Alcover, Apr 23 2013, after Simon Plouffe *)
Table[3*Binomial[4*n+8,n]/(n+3), {n,0,40}] (* G. C. Greubel, Sep 01 2025 *)

N = 3*66;  x = 'x + O('x^N);
g=serreverse(x-x^4)/x;
gf=g^9;  v=Vec(gf);
vector(#v\3,n,v[3*n-2])
/* Joerg Arndt, Apr 23 2013 */

def A006634(n): return 3*binomial(4*(n+2),n)//(n+3)
print([A006634(n) for n in range(41)]) # G. C. Greubel, Sep 01 2025

G.f.: hypergeom([9/4, 5/2, 11/4, 3], [10/3, 11/3, 4], 256/27*x). - Simon Plouffe, Master's Thesis, UQAM, 1992
G.f.: g^9 where g = 1+x*g^4 is the g.f. of A002293. - Mark van Hoeij, Apr 22 2013
From G. C. Greubel, Sep 01 2025: (Start)
a(n) = 3*binomial(4*n+8, n)/(n+3).
E.g.f.: hypergeom([9/4, 5/2, 11/4, 3], [1, 10/3, 11/3, 4], 256*x/27). (End)

More terms from Joerg Arndt, Apr 23 2013

New name by G. C. Greubel, Sep 01 2025

A006633 Expansion of hypergeom([3/2, 7/4, 2, 9/4], [7/3, 8/3, 3], (256/27)*x).

Original entry on oeis.org

1, 6, 39, 272, 1995, 15180, 118755, 949344, 7721604, 63698830, 531697881, 4482448656, 38111876530, 326439471960, 2814095259675, 24397023508416, 212579132600076, 1860620845932216, 16351267454243260, 144222309948974400, 1276307560533365955, 11329053395044653180

Offset: 0

Simon Plouffe

nonn

From generalized Catalan numbers.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Paolo Xausa, Table of n, a(n) for n = 0..1000
H. M. Finucan, Some decompositions of generalized Catalan numbers, pp. 275-293 of Combinatorial Mathematics IX. Proc. Ninth Australian Conference (Brisbane, August 1981). Ed. E. J. Billington, S. Oates-Williams and A. P. Street. Lecture Notes Math., 952. Springer-Verlag, 1982.
Simon Plouffe, Approximations of generating functions and a few conjectures, Master's Thesis, arXiv:0911.4975 [math.NT], 2009.

Cf. A006631, A006632, A006634, A006635, A000027, A000260.

gf := hypergeom([3/2, 7/4, 2, 9/4], [7/3, 8/3, 3], (256/27)*x):
ser := series(gf, x, 22): seq(coeff(ser, x, n), n = 0..21); # Peter Luschny, Feb 22 2024

A006633[n_] := 2*Binomial[4*n+5, n]/(n+2);
Array[A006633, 25, 0] (* Paolo Xausa, Feb 25 2024 *)

O.g.f.: hypergeom_4F3([3/2, 7/4, 2, 9/4], [7/3, 8/3, 3], (256/27)*x). - Simon Plouffe, Master's Thesis, UQAM 1992
a(n) = 2*binomial(4*n + 5, n) / (n+2). - Bruno Berselli, Jan 18 2014
a(n) = (n+1) * A000260(n+1). - F. Chapoton, Feb 22 2024

New name by using a formula from the author by Peter Luschny, Feb 24 2024

A370057 a(n) = 3(4n+2)!/(3*n+3)!.

Original entry on oeis.org

1, 3, 30, 546, 14688, 526680, 23680800, 1282554000, 81339793920, 5915366392320, 485415660038400, 44376781223174400, 4473125162795520000, 492902545595556096000, 58949616073242166272000, 7605168496387089788160000, 1052810955815818170875904000

Offset: 0

Seiichi Manyama, Feb 08 2024

nonn

Harvey P. Dale, Table of n, a(n) for n = 0..327

Cf. A365340, A370056, A370058.

Cf. A000142, A006632.

Table[(3(4n+2)!)/(3n+3)!,{n,0,20}] (* Harvey P. Dale, Feb 15 2025 *)

PARI
```
a(n) = 3*(4*n+2)!/(3*n+3)!;
```

E.g.f.: exp( 3/4 * Sum_{k>=1} binomial(4*k,k) * x^k/k ).
a(n) = A000142(n) * A006632(n+1).
D-finite with recurrence 3*(3*n+2)*(3*n+1)*(n+1)*a(n) -8*n*(4*n+1)*(2*n+1)*(4*n-1)*a(n-1)=0. - R. J. Mathar, Feb 22 2024
From Seiichi Manyama, Aug 31 2024: (Start)
E.g.f. satisfies A(x) = 1/(1 - x*A(x))^3.
a(n) = 3 * Sum_{k=0..n} (3*n+3)^(k-1) * |Stirling1(n,k)|. (End)

A380515 Expansion of e.g.f. exp(xG(x)^3) where G(x) = 1 + xG(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 7, 109, 2689, 91261, 3950191, 208064137, 12917499169, 923765042809, 74780847503191, 6760168138392901, 675023676995501857, 73787463232202560309, 8763902701210982610559, 1123850728979698205132641, 154757223522414820829369281, 22775744033825102490806751217

Offset: 0

Seiichi Manyama, Jan 26 2025

nonn

Cf. A380513, A380514, A380516.
Cf. A080893, A380511.
Cf. A091695, A250917, A380512.
Cf. A002293, A006632, A370057, A382059.

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

a(n) = 3 * n! * Sum_{k=0..n-1} binomial(3*n+k,k)/((3*n+k) * (n-k-1)!) for n > 0.
a(n) = U(1-n, 2-4*n, 1), where U is the Tricomi confluent hypergeometric function. - Stefano Spezia, Jan 26 2025
E.g.f.: exp( Series_Reversion( x*(1-x)^3 ) ). - Seiichi Manyama, Mar 15 2025

A233658 7binomial(4n + 7, n)/(4*n + 7).

Original entry on oeis.org

1, 7, 49, 357, 2695, 20930, 166257, 1344904, 11042724, 91801255, 771201431, 6536904290, 55838330730, 480197194260, 4154140621425, 36126361733616, 315647802951628, 2769544822393356, 24392874398953060, 215582307059144025, 1911286446370861455, 16993580092566979770, 151491588134469616215

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=4, 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, A002293, A069271, A006632, A196678, A006633, A233666, A006634, A233667.

[7*Binomial(4*n+7,n)/(4*n+7): n in [0..30]];

Table[7 Binomial[4 n + 7, n]/(4 n + 7), {n, 0, 30}]

PARI
```
a(n) = 7*binomial(4*n+7,n)/(4*n+7);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(4/7))^7+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=4, r=7.
D-finite with recurrence 3*(3*n+5)*(3*n+7)*(n+2)*a(n) -(n+1)*(661*n^2+1301*n+558)*a(n-1) +120*(4*n+1)*(2*n+1)*(4*n-1)*a(n-2)=0. - R. J. Mathar, Nov 22 2024
D-finite with recurrence 3*n*(3*n+5)*(3*n+7)*(n+2)*a(n) -8*(4*n+5)*(2*n+3)*(4*n+3)*(n+1)*a(n-1)=0. - R. J. Mathar, Nov 22 2024

A233667 a(n) = 5binomial(4n+10,n)/(2*n+5).

Original entry on oeis.org

1, 10, 85, 700, 5750, 47502, 395560, 3321120, 28102425, 239503550, 2054455634, 17726454200, 153757722300, 1340045361750, 11729338225200, 103068670351552, 908923976461140, 8041606944709800, 71359997110169625, 634978885837495500, 5664526697522326590

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=4, 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, A002293, A069271, A006632, A196678, A006633, A233658, A233666, A006634, A006635.

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

Table[5 Binomial[4 n + 10, n]/(2 n + 5), {n, 0, 30}]

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

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(2/5))^10+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=4, r=10.
From Ilya Gutkovskiy, Sep 14 2018: (Start)
E.g.f.: 4F4(5/2,11/4,3,13/4; 1,11/3,4,13/3; 256*x/27).
a(n) ~ 5*2^(8*n+39/2)/(sqrt(Pi)*3^(3*n+21/2)*n^(3/2)). (End)

A371483 G.f. A(x) satisfies A(x) = 1 / (1 - x*A(x) / (1-x))^3.

Original entry on oeis.org

1, 3, 18, 124, 933, 7446, 61943, 531348, 4666425, 41751325, 379230711, 3487769871, 32414437521, 303950138604, 2872137458010, 27322233357964, 261446381792670, 2514851398148595, 24303030755342128, 235841264063844258, 2297278004837062317

Offset: 0

Seiichi Manyama, Mar 25 2024

nonn

Cf. A002212, A270386, A371486.

Cf. A006632, A349331.

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

a(n) = Sum_{k=0..n} binomial(n-1,n-k) * binomial(4*k+2,k)/(k+1).
G.f.: A(x) = B(x/(1-x)), where B(x) = (1/x) * Series_Reversion( x*(1-x)^3 ).
G.f.: A(x) = B(x)^3 where B(x) is the g.f. of A349331.

A386565 a(n) = Sum_{k=0..n-1} binomial(4k-1,k) binomial(4n-4k,n-k-1).

Original entry on oeis.org

0, 1, 11, 111, 1091, 10596, 102237, 982458, 9415539, 90063180, 860278156, 8208539351, 78258171957, 745595635084, 7099714918062, 67574576298276, 642927956583123, 6115089154367484, 58146652079312580, 552769690436583532, 5253812277363417836, 49925987913040522128

Offset: 0

Seiichi Manyama, Jul 26 2025

nonn

			(1/3) * log( Sum_{k>=0} binomial(4*k-1,k)*x^k ) = x + 11*x^2/2 + 37*x^3 + 1091*x^4/4 + 10596*x^5/5 + ...

Cf. A000346, A062236, A386566, A386567.

Cf. A002293, A006632, A308523.

a(n) = sum(k=0, n-1, binomial(4*k-1, k)*binomial(4*n-4*k, n-k-1));

my(N=30, x='x+O('x^N), g=sum(k=0, N, binomial(4*k, k)/(3*k+1)*x^k)); concat(0, Vec(g*(g-1)/(4-3*g)^2))

G.f.: g*(g-1)/(4-3*g)^2 where g=1+x*g^4.
G.f.: g/(1-4*g)^2 where g*(1-g)^3 = x.
L.g.f.: Sum_{k>=1} a(k)*x^k/k = (1/3) * log( Sum_{k>=0} binomial(4*k-1,k)*x^k ).
a(n) = Sum_{k=0..n-1} binomial(4*k-1+l,k) * binomial(4*n-4*k-l,n-k-1) for every real number l.
a(n) = Sum_{k=0..n-1} 3^(n-k-1) * binomial(4*n,k).
a(n) = Sum_{k=0..n-1} 4^(n-k-1) * binomial(3*n+k,k).

A233666 a(n) = 2binomial(4n + 8, n)/(n + 2).

Original entry on oeis.org

1, 8, 60, 456, 3542, 28080, 226548, 1855040, 15380937, 128896456, 1090119316, 9292881360, 79769043900, 688915123680, 5981962494852, 52193342019456, 457367224685012, 4023551800087200, 35521420783728880, 314608026125871720, 2794654131668318430

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=4, 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, A002293, A069271, A006632, A196678, A006633, A233658, A006634, A233667, A006635.

[2*Binomial(4*n+8,n)/(n+2): n in [0..30]]; // Vincenzo Librandi, Dec 14 2013

Table[2/(n + 2) Binomial[4 n + 8, n], {n, 0, 40}] (* Vincenzo Librandi, Dec 14 2013 *)

PARI
```
a(n) = 4*binomial(4*n+8,n)/(n+2);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(1/2))^8+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=4, r=8.
From Ilya Gutkovskiy, Sep 14 2018: (Start)
E.g.f.: 4F4(2,9/4,5/2,11/4; 1,3,10/3,11/3; 256*x/27).
a(n) ~ 2^(8*n+35/2)/(sqrt(Pi)*3^(3*n+17/2)*n^(3/2)). (End)
D-finite with recurrence 3*(3*n+7)*(n+2)*(3*n+8)*a(n) -2*(n+1)*(317*n^2+954*n+709)*a(n-1) +112*(4*n+1)*(2*n+1)*(4*n+3)*a(n-2)=0. - R. J. Mathar, Nov 22 2024

cp's OEIS Frontend

A124753 a(3n+k) = (k+1)*binomial(4n+k, n)/(3n+k+1), where k is n reduced mod 3.

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A006634 a(n) = 3*binomial(4*n+8, n)/(n+3).

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A006633 Expansion of hypergeom([3/2, 7/4, 2, 9/4], [7/3, 8/3, 3], (256/27)*x).

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A370057 a(n) = 3*(4*n+2)!/(3*n+3)!.

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A380515 Expansion of e.g.f. exp(x*G(x)^3) where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.

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A233658 7*binomial(4*n + 7, n)/(4*n + 7).

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A233667 a(n) = 5*binomial(4*n+10,n)/(2*n+5).

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A006634 a(n) = 3binomial(4n+8, n)/(n+3).

A370057 a(n) = 3(4n+2)!/(3*n+3)!.

A380515 Expansion of e.g.f. exp(xG(x)^3) where G(x) = 1 + xG(x)^4 is the g.f. of A002293.

A233658 7binomial(4n + 7, n)/(4*n + 7).

A233667 a(n) = 5binomial(4n+10,n)/(2*n+5).

A386565 a(n) = Sum_{k=0..n-1} binomial(4k-1,k) binomial(4n-4k,n-k-1).

A233666 a(n) = 2binomial(4n + 8, n)/(n + 2).