A069271 -id:A069271 - 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

A233834 a(n) = 5binomial(7n+5,n)/(7*n+5).

Original entry on oeis.org

1, 5, 45, 500, 6200, 82251, 1142295, 16398200, 241379325, 3623534200, 55262073757, 853814730600, 13335836817420, 210225027967325, 3340362288091500, 53443628421286320, 860246972339613855, 13921016318025200505, 226352372251889455000, 3696160728052814340000

Offset: 0

Tim Fulford, Dec 16 2013

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

Cf. A000108, A002296, A233832, A233833, A143547, A130565, A233835, A233907, A233908, A002296, A069271, A118970, A212073, A234465, A234510, A234571, A235339.

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

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

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

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

G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, where p = 7, r = 5.

O.g.f. A(x) = 1/x * series reversion (x/C(x)^5), where C(x) is the o.g.f. for the Catalan numbers A000108. A(x)^(1/5) is the o.g.f. for A002296. - Peter Bala, Oct 14 2015

A235339 a(n) = 9binomial(11n+9,n)/(11*n+9).

Original entry on oeis.org

1, 9, 135, 2460, 49725, 1072197, 24163146, 562311720, 13409091540, 325949656825, 8046743477058, 201198155083200, 5084704634041305, 129673310477725350, 3332952595603387800, 86250038091202771344, 2245329811618166111985

Offset: 0

Tim Fulford, Jan 06 2014

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p = 11, r = 9.

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906 [math.CO], 2007; 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. A230388, A234868, A234869, A234870, A234871, A234872, A234873, A235338, A235340, A069271, A118970, A212073, A230388, A233834, A234465, A234510, A234571.

[9*Binomial(11*n+9, n)/(11*n+9): n in [0..30]];

Table[9 Binomial[11 n + 9, n]/(11 n + 9), {n, 0, 30}]

PARI
```
a(n) = 9*binomial(11*n+9,n)/(11*n+9);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(11/9))^9+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, here p = 11, r = 9.

O.g.f. A(x) = 1/x * series reversion (x/C(x)^9), where C(x) is the o.g.f. for the Catalan numbers A000108. A(x)^(1/9) is the o.g.f. for A230388. - Peter Bala, Oct 14 2015

A215715 G.f. satisfies A(x) = (1 + xA(x)^2) (1 + x*A(x)^4).

Original entry on oeis.org

1, 2, 13, 118, 1242, 14227, 172177, 2165732, 28032668, 370944717, 4995412647, 68239105203, 943278064473, 13169938895473, 185453340189492, 2630813161415976, 37561512615867450, 539336703889993006, 7783290731579783544, 112828761898680983141, 1642222504807143423470

Offset: 0

Paul D. Hanna, Aug 21 2012

nonn

More generally, for fixed parameters p, q, r, and s, if F(x) satisfies:
F(x) = (1 + x^r*F(x)^(p+1)) * (1 + x^(r+s)*F(x)^(p+q+1)), then
F(x) = exp( Sum_{n>=1} x^(n*r)*F(x)^(n*p)/n * [Sum_{k=0..n} C(n,k)^2 * x^(k*s)*F(x)^(k*q)] ).
The radius of convergence of g.f. A(x) is r = 0.06368546004073732405169450... with A(r) = 1.3960637117611795281240000742797488619448782873... where y=A(r) satisfies 6*y^5 + 17*y^4 - 46*y^3 + 16*y^2 + 4*y - 8 = 0.

			G.f.: A(x) = 1 + 2*x + 13*x^2 + 118*x^3 + 1242*x^4 + 14227*x^5 + ...
Related expansions.
A(x)^2 = 1 + 4*x + 30*x^2 + 288*x^3 + 3125*x^4 + 36490*x^5 + ...
A(x)^4 = 1 + 8*x + 76*x^2 + 816*x^3 + 9454*x^4 + 115260*x^5 + ...
A(x)^6 = 1 + 12*x + 138*x^2 + 1648*x^3 + 20427*x^4 + 260934*x^5 + ...
where A(x) = 1 + x*(A(x)^2 + A(x)^4) + x^2*A(x)^6.
The logarithm of the g.f. equals the series:
log(A(x)) = (1 + A(x)^2)*x*A(x) + (1 + 2^2*A(x)^2 + A(x)^4)*x^2*A(x)^2/2 +
(1 + 3^2*A(x)^2 + 3^2*A(x)^4 + A(x)^6)*x^3*A(x)^3/3 +
(1 + 4^2*A(x)^2 + 6^2*A(x)^4 + 4^2*A(x)^6 + A(x)^8)*x^4*A(x)^4/4 +
(1 + 5^2*A(x)^2 + 10^2*A(x)^4 + 10^2*A(x)^6 + 5^2*A(x)^8 + A(x)^10)*x^5*A(x)^5/5 + ...
Explicitly,
log(A(x)) = 2*x + 22*x^2/2 + 284*x^3/3 + 3878*x^4/4 + 54607*x^5/5 + 784144*x^6/6 + 11414265*x^7/7 + 167819014*x^8/8 + ...

G. C. Greubel, Table of n, a(n) for n = 0..825
Vaclav Kotesovec, Recurrence

Cf. A198953.

Cf. A007863, A069271, A073157, A215654.

CoefficientList[Sqrt[1/x*InverseSeries[Series[x*(1+Sqrt[1-4*x*(1+x)^2])^2/(4*(1+x)^2),{x,0,20}],x]],x] (* Vaclav Kotesovec, Sep 17 2013 *)

{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=(1 + x*A^2)*(1 + x*A^4)); polcoeff(A, n)}
for(n=0,31,print1(a(n),", "))

{a(n)=polcoeff( sqrt((1/x)*serreverse( x*(1 + sqrt(1 - 4*x*(1+x)^2 +x*O(x^n)))^2/(4*(1+x)^2))), n)}

{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2*A^(2*j))*x^m*A^m/m))); polcoeff(A, n)}

{a(n)=local(A=1+x+x*O(x^n)); for(i=1, n, A=exp(sum(m=1, n, sum(j=0, m, binomial(m, j)^2/A^(2*j))*x^m*A^(3*m)/m))); polcoeff(A, n)}

G.f. A(x) satisfies:
(1) A(x) = sqrt( (1/x)*Series_Reversion( x*(1 + sqrt(1 - 4*x*(1+x)^2))^2/(4*(1+x)^2) ) ).
(2) A(x) = exp( Sum_{n>=1} x^n*A(x)^n/n * Sum_{k=0..n} C(n,k)^2 * A(x)^(2*k) ).
(3) A(x) = exp( Sum_{n>=1} x^n*A(x)^(3*n)/n * Sum_{k=0..n} C(n,k)^2 / A(x)^(2*k) ).
The formal inverse of the g.f. A(x) is (sqrt(x^4 + 4*x^3 - 2*x^2 + 1) - (1+x^2))/(2*x^4).
a(n) ~ c*d^n/(sqrt(Pi)*n^(3/2)), where d = 15.70217125479403872... is the root of the equation -1024 - 3840*d + 26368*d^2 - 58644*d^3 + 1933*d^4 + 108*d^5 = 0 and c = 0.320114409... - Vaclav Kotesovec, Sep 17 2013
a(n) = Sum_{k=0..n} binomial(2*n+2*k+1,k) * binomial(2*n+2*k+1,n-k) / (2*n+2*k+1). - Seiichi Manyama, Jul 18 2023

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)

A364331 G.f. satisfies A(x) = (1 + xA(x)^2) (1 + x*A(x)^5).

Original entry on oeis.org

1, 2, 15, 163, 2070, 28698, 421015, 6425644, 100977137, 1622885389, 26551709946, 440744175801, 7404449354076, 125657625548824, 2150963575012295, 37094953102567208, 643904274979347286, 11241232087809137759, 197247501440314516840, 3476787208220672891388, 61533794803235280779261

Offset: 0

Seiichi Manyama, Jul 18 2023

Cf. A007863, A069271, A073157, A215654, A215715, A364333.
Cf. A215623, A215624, A239108, A364335, A364338.
Cf. A200719.

A364331 := proc(n)
    add( binomial(2*n+3*k+1,k) * binomial(2*n+3*k+1,n-k)/(2*n+3*k+1),k=0..n) ;
end proc:
seq(A364331(n),n=0..70); # R. J. Mathar, Jul 25 2023

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

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

x/series_reversion(x*A(x)) = 1 + 2*x + 11*x^2 + 89*x^3 + 836*x^4 + ..., the g.f. of A215623. - Peter Bala, Sep 08 2024

A381772 Expansion of ( (1/x) * Series_Reversion( x/((1+x) * C(x))^2 ) )^(1/2), where C(x) is the g.f. of A000108.

Original entry on oeis.org

1, 2, 11, 83, 727, 6940, 70058, 735502, 7949031, 87851819, 988307647, 11279719247, 130286197186, 1520108988221, 17889102534329, 212095541328931, 2531001870925559, 30376237591559863, 366417240105654587, 4440000077166319993, 54020150448778625847, 659665548217188211288

Offset: 0

Seiichi Manyama, Mar 07 2025

nonn

Cf. A054727, A060941, A381773, A381774, A381775.
Cf. A007852, A069271, A381780.
Cf. A000108, A274052.

my(N=30, x='x+O('x^N)); Vec((serreverse(x/((1+x)*(1-sqrt(1-4*x))/(2*x))^2)/x)^(1/2))

G.f. A(x) satisfies A(x) = (1 + x*A(x)^2) * C(x*A(x)^2).

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

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

A380514 Expansion of e.g.f. exp(xG(x)^2) where G(x) = 1 + xG(x)^4 is the g.f. of A002293.

Original entry on oeis.org

1, 1, 5, 67, 1537, 50021, 2107021, 108885295, 6665443457, 471522589417, 37843890892021, 3397250515809371, 337267132243022785, 36687625652474612557, 4339368321317331858557, 554467482301151809302151, 76112537023512618262963201, 11170667360636927554290623825, 1745500813880455301486766050917

Offset: 0

Seiichi Manyama, Jan 26 2025

nonn

Cf. A002293, A069271, A380513, A380515, A380516.

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

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

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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A233834 a(n) = 5*binomial(7*n+5,n)/(7*n+5).

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A235339 a(n) = 9*binomial(11*n+9,n)/(11*n+9).

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A215715 G.f. satisfies A(x) = (1 + x*A(x)^2) * (1 + x*A(x)^4).

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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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A233834 a(n) = 5binomial(7n+5,n)/(7*n+5).

A235339 a(n) = 9binomial(11n+9,n)/(11*n+9).

A215715 G.f. satisfies A(x) = (1 + xA(x)^2) (1 + x*A(x)^4).

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

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

A364331 G.f. satisfies A(x) = (1 + xA(x)^2) (1 + x*A(x)^5).

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

A380514 Expansion of e.g.f. exp(xG(x)^2) where G(x) = 1 + xG(x)^4 is the g.f. of A002293.