A002294 -id:A002294 - 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

A349300 G.f. A(x) satisfies: A(x) = 1 / ((1 + x) * (1 - x * A(x)^4)).

Original entry on oeis.org

1, 0, 1, 4, 21, 114, 651, 3844, 23301, 144169, 906866, 5782350, 37289431, 242793439, 1593918916, 10538988984, 70121101825, 469133993094, 3154115695476, 21299373321344, 144402246424591, 982506791975780, 6706724412165956, 45917245477282994

Offset: 0

Ilya Gutkovskiy, Nov 13 2021

Seiichi Manyama, Table of n, a(n) for n = 0..500

Cf. A002294, A005043, A346627, A346665, A349290, A349299, A349301, A349302, A349303.

nmax = 23; A[] = 0; Do[A[x] = 1/((1 + x) (1 - x A[x]^4)) + O[x]^(nmax + 1) // Normal, nmax + 1]; CoefficientList[A[x], x]
Table[Sum[(-1)^(n - k) Binomial[n + 3 k, 4 k] Binomial[5 k, k]/(4 k + 1), {k, 0, n}], {n, 0, 23}]

a(n) = sum(k=0, n, (-1)^(n-k) * binomial(n+3*k,4*k) * binomial(5*k,k) / (4*k+1)); \\ Michel Marcus, Nov 14 2021

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n+3*k,4*k) * binomial(5*k,k) / (4*k+1).
a(n) = (-1)^5*F([1/5, 2/5, 3/5, 4/5, (1+n)/3, (2+n)/3, (3+n)/3, -n], [1/4, 1/2, 1/2, 3/4, 3/4, 1, 5/4], 3^3*5^5/2^16), where F is the generalized hypergeometric function. - Stefano Spezia, Nov 14 2021
a(n) ~ sqrt(1 - 3*r) / (2 * 5^(3/4) * sqrt(2*Pi*(1+r)) * n^(3/2) * r^(n + 1/4)), where r = 0.136824361675510443450981569282313811786270109272790613523286... is the root of the equation 5^5 * r = 4^4 * (1+r)^4. - Vaclav Kotesovec, Nov 14 2021
From Peter Bala, Jun 02 2024: (Start)
A(x) = 1/(1 + x)*F(x/(1 + x)^4), where F(x) = Sum_{n >= 0} A002294(n)*x^n.
A(x) = 1/(1 + x) + x*A(x)^5. (End)

A371739 a(n) = Sum_{k=0..n} binomial(5*n,k).

Original entry on oeis.org

1, 6, 56, 576, 6196, 68406, 768212, 8731848, 100146724, 1156626990, 13432735556, 156713948672, 1835237017324, 21560768699762, 253994850228896, 2999267652451776, 35490014668470052, 420718526924212654, 4995548847105422048, 59402743684137281920

Offset: 0

Seiichi Manyama, Apr 05 2024

Cf. A032443, A066380, A066381.

Cf. A002294, A163455, A386699.

Table[32^n - Binomial[5*n, 1+n] * Hypergeometric2F1[1, 1 - 4*n, 2+n, -1], {n, 0, 20}] (* Vaclav Kotesovec, Apr 05 2024 *)

PARI
```
a(n) = sum(k=0, n, binomial(5*n, k));
```

a(n) = [x^n] 1/((1-2*x) * (1-x)^(4*n)).
a(n) ~ 5^(5*n + 1/2) / (3*sqrt(Pi*n) * 2^(8*n - 1/2)). - Vaclav Kotesovec, Apr 05 2024
a(n) = Sum_{k=0..floor(n/2)} binomial(5*n+1,n-2*k). - Seiichi Manyama, Apr 09 2024
a(n) = binomial(1+5*n, n)*hypergeom([1, (1-n)/2, -n/2], [1+2*n, 3/2+2*n], 1). - Stefano Spezia, Apr 09 2024
a(n) = Sum_{k=0..n} 2^(n-k) * binomial(4*n+k-1,k). - Seiichi Manyama, Jul 30 2025
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(5*n,k) * binomial(5*n-k-1,n-k). - Seiichi Manyama, Aug 08 2025
G.f.: g/((2-g) * (5-4*g)) where g = 1+x*g^5 is the g.f. of A002294. - Seiichi Manyama, Aug 12 2025
G.f.: 1/(1 - x*g^4*(10-4*g)) where g = 1+x*g^5 is the g.f. of A002294. - Seiichi Manyama, Aug 17 2025

A062986 Coefficient array for certain polynomials N(5; k,x) (rising powers in x).

Original entry on oeis.org

1, 5, -10, 10, -5, 1, 35, -170, 415, -629, 630, -420, 180, -45, 5, 285, -2315, 9381, -24395, 44625, -59880, 60015, -45040, 25025, -10010, 2730, -455, 35, 2530, -29379, 169405, -633675, 1703700, -3467145, 5497640, -6903325

Offset: 0

Wolfdieter Lang, Jul 12 2001

The g.f. for the sequence of column r=4*k+j, k >= 0, j=1,2,3,4, of the staircase array A062985(n,r) is N(5; k,x)*(x^(k+1))/(1-x)^(4*k+1+j) with N(5; k,x) := sum(a(k,p)*x^p,p=0..4*k).
The m=0 column gives A002294(k+1). The row sums give A000012 (powers of 1) and (unsigned) A062987.
The sequence of step width of this staircase array is [1,4,4,4,...], i.e. the degree of the row polynomials is [0,4,8,12,...]= A008586.

			{1}; {5,-10,10,-5,1}; {35,-170,415,-629,630,-420,180,-45,5}; ...; N(5; 1,x)= 5-10*x+10*x^2-5*x^3+x^4 = (1-(1-x)^5)/x.

A062991, A062746, A062751.

a(k, p) := [x^p]N(5; k, x) with N(5; k, x)=(N(5; k-1, x)- A002294(k)*(1-x)^(4*k+1))/x, N(5; 0, x) := 1.

a(n, k)= a(n-1, k+1)+((-1)^k)*binomial(4*n+1, k+1)*binomial(5*n+1, n)/(5*n+1) if k=0, .., (4*n-5); a(n, k)= ((-1)^k)*binomial(4*n+1, k+1)*binomial(5*n+1, n)/(5*n+1) if k=(4*n-4), ..., 4*n; else 0.

A110448 G.f.: A(x) = exp( Sum_{n>=1} A056045(n)/n*x^n ), where A056045(n) = Sum_{d|n} binomial(n,d).

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 18, 23, 49, 73, 145, 194, 474, 611, 1331, 2027, 4393, 5919, 14736, 19415, 46487, 68504, 156618, 212055, 560380, 739165, 1833012, 2657837, 6513367, 8743208, 23649777, 31140300, 81276046, 114962333, 293600318, 391926154

Offset: 0

Paul D. Hanna, Jul 20 2005, Nov 10 2007

nonn

			G.f.: A(x) = 1 + x + 2*x^2 + 3*x^3 + 6*x^4 + 8*x^5 + 18*x^6 +...
where A(x) = exp( Sum_{n>=1} A056045(n)/n*x^n ), or
A(x) = exp(x + 3/2*x^2 + 4/3*x^3 + 11/4*x^4 + 6/5*x^5 +...).
The g.f. can also be expressed as the product:
A(x) = 1/(1-x)*G000108(x^2)*G001764(x^3)*G002293(x^4)*G002294(x^5)*...
where the functions are g.f.s of well-known sequences:
G000108(x) = 1 + x*G000108(x)^2 = g.f. of A000108 ;
G001764(x) = 1 + x*G001764(x)^3 = g.f. of A001764 ;
G002293(x) = 1 + x*G002293(x)^4 = g.f. of A002293 ;
G002294(x) = 1 + x*G002294(x)^5 = g.f. of A002294 ; etc.

Seiichi Manyama, Table of n, a(n) for n = 0..3337

Cf. A056045, A174461, A206290.

Cf. A000108, A001764, A002293, A002294, A002295.

{a(n)=polcoeff(exp(x*Ser(vector(n,m, sumdiv(m,d,binomial(m,d))/m))+x*O(x^n)),n)}

{a(n)=polcoeff(prod(m=1,n,1/x*serreverse(x/(1+x^m +x*O(x^n)))),n)}

G.f.: A(x) = Product_{n>=1} (1/x)*Series_Reversion( x/(1 + x^n) ); equivalently, G.f.: A(x) = Product_{n>=1} G(x^n,n) where G(x,n) = 1 + x*G(x,n)^n.

a(n) ~ c * 2^n / n^(3/2), where c = 2.8176325363130737043447... if n is even and c = 1.784372019603712867208... if n is odd. - Vaclav Kotesovec, Jan 15 2019

A113207 Coefficients of inverse of Poincaré series [or Poincare series] of the 5-Gonal operad.

Original entry on oeis.org

1, 5, 38, 347, 3507, 37788, 425490, 4947239, 58944743, 715930085, 8831390152, 110340491380, 1393446215956, 17758187064360, 228091606247322, 2949707761133535, 38374765966463775, 501891882459954495, 6595106960772794310, 87030030852121334835, 1152846885317408648715

Offset: 1

N. J. A. Sloane, Jan 07 2006

nonn

Leroux asks: Is there a combinatorial interpretation for these numbers?

Those are the coefficients of the series reverse of the Poincaré series of the 5-Gonal operad, and not of the 5-Tetra operad. The sequence for the 5-Tetra operad is well-known and is A002294. - Paul Laubie, Nov 08 2023

Ph. Leroux, A simple symmetry generating operads related to rooted planar m-ary trees and polygonal numbers, arXiv:math/0512437 [math.CO], 2005.
Ph. Leroux, A simple symmetry generating operads related to rooted planar m-ary trees and polygonal numbers, J. Integer Seqs., 10 (2007), #07.4.7.

R.=PowerSeriesRing(QQ); (-(2*t^2-t)/(1+t)^3).reverse().list()[1:] # Paul Laubie, Nov 08 2023

G.f.: series reversion of -(2*t^2-t)/(1+t)^3. - Paul Laubie, Nov 08 2023

New offset, name corrected and more terms from Paul Laubie, Nov 08 2023

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)

A334719 a(n) is the total number of down-steps after the final up-step in all 4-Dyck paths of length 5n (n up-steps and 4n down-steps).

Original entry on oeis.org

0, 4, 30, 250, 2245, 21221, 208129, 2098565, 21619910, 226593015, 2408424760, 25899375645, 281273231985, 3080585212120, 33986840371400, 377364606387005, 4213620859310140, 47284625533425750, 532996618440511710, 6032169040263819485, 68517222947120776290

Offset: 0

Andrei Asinowski, May 08 2020

A 4-Dyck path is a lattice path with steps U = (1, 4), d = (1, -1) that starts at (0,0), stays (weakly) above the x-axis, and ends at the x-axis.

			For n = 2, the a(2) = 30 is the total number of down-steps after the last up-step in UddddUdddd, UdddUddddd, UddUdddddd, UdUddddddd, UUdddddddd (thus, 4 + 5 + 6 + 7 + 8).

Stefano Spezia, Table of n, a(n) for n = 0..900
Andrei Asinowski, Benjamin Hackl, and Sarah J. Selkirk, Down-step statistics in generalized Dyck paths, arXiv:2007.15562 [math.CO], 2020.

First order differences of A002294. Cf. A062985.

Cf. A334682 (similar for 3-Dyck paths).

b:= proc(x, y) option remember; `if`(x=y, x,
     `if`(y+40, b(x-1, y-1), 0))
    end:
a:= n-> b(5*n, 0):
seq(a(n), n=0..20);  # Alois P. Heinz, May 09 2020
# second Maple program:
a:= proc(n) option remember; `if`(n<2, 4*n, (5*(5*n-4)*
      (5*n-3)*(5*n-2)*(5*n-1)*n*(2869*n^3+5354*n^2+3239*n+634)*
       a(n-1))/(8*(n-1)*(4*n+3)*(2*n+1)*(4*n+5)*(n+1)*
       (2869*n^3-3253*n^2+1138*n-120)))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, May 09 2020

a[n_] := Binomial[5*n + 6, n + 1]/(5*n + 6) - Binomial[5*n + 1, n]/(5*n + 1); Array[a, 21, 0] (* Amiram Eldar, May 13 2020 *)

a(n) = {binomial(5*(n+1)+1, n+1)/(5*(n+1)+1) - binomial(5*n+1, n)/(5*n+1)} \\ Andrew Howroyd, May 08 2020

[binomial(5*(n + 1) + 1, n + 1)/(5*(n + 1) + 1) - binomial(5*n + 1, n)/(5*n + 1) for n in srange(30)] # Benjamin Hackl, May 13 2020

a(n) = binomial(5*(n+1)+1, n+1)/(5*(n+1)+1) - binomial(5*n+1, n)/(5*n+1).
a(n) = A062985(n+1, 4*n-1).
G.f.: ((1 - x)*HypergeometricPFQ([1/5, 2/5, 3/5, 4/5], [1/2, 3/4, 5/4], 3125*x/256) - 1)/x. - Stefano Spezia, Apr 25 2023

cp's OEIS Frontend

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

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A349300 G.f. A(x) satisfies: A(x) = 1 / ((1 + x) * (1 - x * A(x)^4)).

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A371739 a(n) = Sum_{k=0..n} binomial(5*n,k).

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A062986 Coefficient array for certain polynomials N(5; k,x) (rising powers in x).

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A110448 G.f.: A(x) = exp( Sum_{n>=1} A056045(n)/n*x^n ), where A056045(n) = Sum_{d|n} binomial(n,d).

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A113207 Coefficients of inverse of Poincaré series [or Poincare series] of the 5-Gonal operad.

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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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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).

A334719 a(n) is the total number of down-steps after the final up-step in all 4-Dyck paths of length 5n (n up-steps and 4n down-steps).