A130564 -id:A130564 - cp's OEIS frontend

A062994 Eighth column of triangle A062993 (without leading zeros). A Pfaff-Fuss or 9-Raney sequence.

1, 1, 9, 117, 1785, 29799, 527085, 9706503, 184138713, 3573805950, 70625252863, 1416298046436, 28748759731965, 589546754316126, 12195537924351375, 254184908607118800, 5332692942907262361

Offset: 0

Wolfdieter Lang, Jul 12 2001

See Graham et al., Hilton and Pedersen, Hoggat and Bicknell, Frey and Sellers references given in A062993.
Essentially the same as A059967. a(n), n>=1, enumerates 9-ary trees (rooted, ordered, incomplete) with n vertices (including the root).
These numbers appear in a formula on p. 24 of Gross et al. for b = -2 or 4. For b = -1 or 3, see A002293.- Tom Copeland, Dec 24 2019
This is instance k = 9 of the generalized Catalan family {C(k, n)}_{n>=0} given in a comment of A130564. - Wolfdieter Lang, Feb 05 2024

			There are a(2)=9 9-ary trees (vertex degree <=9 and 9 possible branchings) with 2 vertices (one of them the root). Adding one more branch (one more vertex) to these 9 trees yields 9*9 + binomial(9,2) = 117 = a(3) such trees.

G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, Heidelberg, New York, 2 vols., 1972, Vol. 1, problem. 211, p. 146 with solution on p. 348.

Harry J. Smith, Table of n, a(n) for n = 0..100
M. Gross, P. Hacking, S. Keel, and M. Kontsevich, Canonical bases for cluster algebras, arXiv preprint arXiv:1411.1394 [math.AG], 2016.

Column k=8 of A070914.

Cf. A000108, A001764, A002293, A002294, A002295, A002296, A007556, A059968, A130564.

seq(binomial(9*k+1,k)/(8*k+1),k=0..30);
n:=30: G:=series(RootOf(g = 1+x*g^9, g),x=0,n+1): seq(coeff(G,x,k),k=0..n); # Robert FERREOL, Apr 01 2015

Table[Binomial[9n,n]/(8n+1),{n,0,30}] (* Harvey P. Dale, Oct 28 2012 *)

{ for (n=0, 100, write("b062994.txt", n, " ", binomial(9*n, n)/(8*n + 1)) ) } \\ Harry J. Smith, Aug 15 2009

a(n) = A062993(n+7, 7) = binomial(9*n, n)/(8*n+1).
G.f.: RootOf((_Z^9)*x-_Z+1) (Maple notation, from ECS, see links for A007556).
Recurrence: a(0) = 1; a(n) = Sum_{i1+i2+..+i9=n-1} a(i1)*a(i2)*...*a(i9) for n>=1. - Robert FERREOL, Apr 01 2015
From Ilya Gutkovskiy, Jan 16 2017: (Start)
O.g.f.: 8F7(1/9,2/9,1/3,4/9,5/9,2/3,7/9,8/9; 1/4,3/8,1/2,5/8,3/4,7/8,9/8; 387420489*x/16777216).
E.g.f.: 8F8(1/9,2/9,1/3,4/9,5/9,2/3,7/9,8/9; 1/4,3/8,1/2,5/8,3/4,7/8,1,9/8; 387420489*x/16777216).
a(n) ~ 3^(18*n+1)/(sqrt(Pi)*2^(24*n+5)*n^(3/2)). (End)
D-finite with recurrence: 128*n*(8*n-5)*(4*n-1)*(8*n+1)*(2*n-1)*(8*n-1)*(4*n-3)*(8*n-3)*a(n) -81*(9*n-7)*(9*n-5)*(3*n-1)*(9*n-1)*(9*n-8)*(3*n-2)*(9*n-4)*(9*n-2)*a(n-1)=0. - R. J. Mathar, Feb 20 2020
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^17). - Seiichi Manyama, Jun 16 2025

9-ary tree comments and Pólya and G. Szegő reference from Wolfdieter Lang, Sep 14 2007

A230388 a(n) = binomial(11n+1,n)/(11n+1).

Original entry on oeis.org

1, 1, 11, 176, 3311, 68211, 1489488, 33870540, 793542167, 19022318084, 464333035881, 11502251937176, 288417894029200, 7306488667126803, 186719056586568660, 4807757550367267056, 124609430032127192295, 3248403420844673986345

Offset: 0

Tim Fulford, Jan 01 2014

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=11, r=1. Interesting property when r=1, a(n+1,p,1) = a(n,p,p) for n>=0.

This is also instance k = 10 of the generalized Catalan family {C(k, n)}A130564.%20-%20_Wolfdieter%20Lang">{n>=0} given in a comment of A130564. - _Wolfdieter Lang, Feb 05 2024

Vincenzo Librandi, Table of n, a(n) for n = 0..200
J-C. Aval, Multivariate Fuss-Catalan Numbers, arXiv:0711.0906 [math.CO], 2008.
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.
J.-C. Novelli, J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv preprint arXiv:1403.5962, 2014
Wikipedia, Fuss-Catalan number

Cf. A234868-A234873, A130564.

[Binomial(11*n+1,n)/(11*n+1): n in [0..30]]; // Vincenzo Librandi, Jan 01 2014

seq(binomial(11*k+1,k)/(11*k+1),k=0..30); # Robert FERREOL, Apr 01 2015
n:=30:G:=series(RootOf(g = 1+x*g^11, g),x=0,n+1):seq(coeff(G,x,k),k=0..n); # Robert FERREOL, Apr 01 2015

Table[Binomial[11 n + 1, n]/(11 n + 1), {n, 0, 30}] (* Vincenzo Librandi, Jan 01 2014 *)

PARI
```
a(n) = binomial(11*n+1,n)/(11*n+1);
```

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

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, with p=11, r=1.
From Robert FERREOL, Apr 01 2015: (Start)
a(n) = binomial(11*n,n)/(10*n+1) = A062993(n+9, 9).
a(0) = 1; a(n) = Sum_{i1+i2+..i11=n-1} a(i1)*a(i2)*...*a(i11) for n>=1.
(End)
O.g.f.: hypergeometric([1,...,10]/11,[2,...,9,11]/10,(11^11/10^10)*x). For the e.g.f. put an extra  1 = 10/10 into the second part. - Wolfdieter Lang, Feb 05 2024
a(n) ~ (11^11/10^10)^n*sqrt(11/(2*Pi*(10*n)^3)). - Robert A. Russell, Jul 15 2024
G.f. A(x) satisfies A(x) = 1/A(-x*A(x)^21). - Seiichi Manyama, Jun 16 2025

A234466 a(n) = 7binomial(8n+7,n)/(8*n+7).

Original entry on oeis.org

1, 7, 77, 1015, 14763, 228459, 3689595, 61474519, 1048927880, 18236463245, 321899509386, 5753527081211, 103922382296180, 1893943017506925, 34783258504651434, 643111366544129175, 11960812088346090200, 223614812152492437432, 4200107505573406222425

Offset: 0

Tim Fulford, Dec 26 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=8, 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.
Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.

Cf. A000108, A007556, A118971, A130564, A234461, A234462, A234463, A234464, A234465, A234467, A230390.

[7*Binomial(8*n+7, n)/(8*n+7): n in [0..30]]; // Vincenzo Librandi, Dec 26 2013

Table[7 Binomial[8 n + 7, n]/(8 n + 7), {n, 0, 40}] (* Vincenzo Librandi, Dec 26 2013 *)

PARI
```
a(n) = 7*binomial(8*n+7,n)/(8*n+7);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(8/7))^7+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=8, r=7.
E.g.f.: hypergeom([7, 9, 10, 11, 12, 13, 14]/8, [8, 9, 10, 11, 12, 13, 14]/7, (8^8/7^7)*x). Cf.: Ilya Gutkovskiy in A118971. - Wolfdieter Lang, Feb 06 2020
D-finite with recurrence: +7*(7*n+4)*(7*n+1)*(7*n+5)*(7*n+2)*(7*n+6)*(7*n+3)*(n+1)*a(n) -128*(8*n+3)*(4*n+3)*(8*n+1)*(2*n+1)*(8*n-1)*(4*n+1)*(8*n+5)*a(n-1)=0. - R. J. Mathar, Feb 21 2020
From Wolfdieter Lang, Feb 15 2024: (Start)
a(n) = binomial(8*n + 6, n+1)/(7*n + 6). This is instance k = 7 of c(k, n+1) given in a comment in A130564.
The compositional inverse of y*(1 - y)^7 is x*G(x), where G is the o.g.f.. That is, G(x)*(1 - x*G(x))^7 = 1. This is equivalent to the formula of the first line above with B = G. Take  A =  B^(1/7) then A*(1 - x*B) = 1 or B*(1 - x*B)^7 = 1.
The o.g.f is G(x) = 8F7([7..14]/8, [8..14]/7; (8^8/7^7)*x) =  (7/(8*x))*(1 - 7F6([-1,1,2,3,4,5,6]/8, [1,2,3,4,5,6]/7; (8^8/7^7)*x)). See the e.g.f. above.(End)

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

Original entry on oeis.org

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

A234513 8binomial(9n+8,n)/(9*n+8).

Original entry on oeis.org

1, 8, 100, 1496, 24682, 433160, 7932196, 149846840, 2898753715, 57135036024, 1143315429776, 23166186450680, 474347963242860, 9799792252101016, 204022381037886400, 4276098770070159096, 90151561242584838605, 1910564646571462338800

Offset: 0

Tim Fulford, Dec 27 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), where p=9, 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.
Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.

Cf. A000108, A118971, A130564, A143554, A234466, A234505, A234506, A234507, A234508, A234509, A234510, A232265.

[8*Binomial(9*n+8, n)/(9*n+8): n in [0..30]]; // Vincenzo Librandi, Dec 28 2013

Table[8 Binomial[9 n + 8, n]/(9 n + 8), {n, 0, 30}] (* Vincenzo Librandi, Dec 28 2013 *)

PARI
```
a(n) = 8*binomial(9*n+8,n)/(9*n+8);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(9/8))^8+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=9, r=8.
From Wolfdieter Lang, Feb 06 2020: (Start)
G.f.: hypergeom([8, 9, ..., 16]/9, [9, 10, ..., 16]/8, (9^9/8^8)*x).
E,g,f.: hypergeom([8, 10, 11, ..., 16]/9, [9, 10,..., 16]/8, (9^9/8^8)*x). Cf. Ilya Gutkovsky in  A118971. (End)
D-finite with recurrence 128*(8*n+3)*(4*n+3)*(8*n+1)*(2*n+1)*(8*n+7)*(4*n+1)*(8*n+5)*(n+1)*a(n) -81*(9*n+2)*(9*n+4)*(3*n+2)*(9*n-1)*(9*n+1)*(3*n+1)*(9*n+5)*(9*n+7)*a(n-1)=0. - R. J. Mathar, Aug 01 2022
From Wolfdieter Lang, Feb 15 2024: (Start)
a(n) = binomial(9*n+7, n+1)/(8*n+7), which is instance k = 8 of c(k, n+1) given in A130564.
The g.f. given above, and called B in the first line above, satisfies B(x)*(1 - x*B(x))^8 = 1. For the analog proof of the equivalence see A234466. x*B(x) is the compositional inverse of y*(1 - y)^8.
Another formula for the g.f. is B(x) =  (8/(9*x))*(1 - 8F7([-1,1,2,3,4,5,6.7]/9, [1,2,3,4,5,6.7]/8; (9^9/8^8)*x)). (End)

A234573 a(n) = 9binomial(10n+9,n)/(10*n+9).

Original entry on oeis.org

1, 9, 126, 2109, 38916, 763686, 15636192, 330237765, 7141879503, 157366449604, 3520256293710, 79735912636302, 1825080422272800, 42148579533938784, 980892581545169496, 22980848343194476245, 541581608172776494554, 12829884648994115426295, 305349921559399354716430

Offset: 0

Tim Fulford, Dec 28 2013

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), where p=10, 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.
Elżbieta Liszewska and Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.
Wojciech Mlotkowski, Fuss-Catalan Numbers in Noncommutative Probability, Docum. Mathm. 15: 939-955.
J. Sawada, J. Sears, A. Trautrim, and A. Williams, Demystifying our Grandparent's De Bruijn Sequences with Concatenation Trees, arXiv:2308.12405 [math.CO], 2023.

Cf. A000108, A059968, A118971, A130564, A234513, A234525, A234526, A234527, A234528, A234529, A234570, A234571, A229963.

[9*Binomial(10*n+9, n)/(10*n+9): n in [0..30]];

Table[9 Binomial[10 n + 9, n]/(10 n + 9), {n, 0, 30}]

PARI
```
a(n) = 9*binomial(10*n+9,n)/(10*n+9);
```

{a(n)=local(B=1); for(i=0, n, B=(1+x*B^(10/9))^9+x*O(x^n)); polcoeff(B, n)}

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=10, r=9.
From Wolfdieter Lang, Feb 06 2020: (Start)
G.f.: hypergeom([9, 10, ..., 18]/10, [10, 11, ..., 18]/9, (10^10/9^9)*x).
E.g.f.: hypergeom([9, 11, 12, ..., 18]/10, [10, 11, ..., 18]/9, (10^10/9^9) * x). Cf. Ilya Gutkovsky in  A118971. (End)
a(n) = binomial(10*n + 8 , n+1)/(9*n + 8) which is instance k = 9 of c(k, n+1) given in a comment in A130564. x*B(x), with the above given g.f. B(x), is the compositional inverse of y*(1 - y)^9, hence  B(x)*(1 - x*B(x))^9 = 1. For another formula for B(x) involving the hypergeometric function 9F8 see the analog formula in A234513. - Wolfdieter Lang, Feb 15 2024

A130565 Member k=6 of a family of generalized Catalan numbers.

Original entry on oeis.org

1, 6, 57, 650, 8184, 109668, 1533939, 22137570, 327203085, 4928006512, 75357373305, 1166880131820, 18259838103852, 288308609783760, 4587430875645660, 73484989079268690, 1184104656043939071

Offset: 1

Wolfdieter Lang, Jul 13 2007

The generalized Catalan numbers C(k,n):= binomial(k*n+1,n)/(k*n+1) become for negative k=-|k|, with |k|>=2, ((-1)^(n-1))*binomial((|k|+1)*n-2,n)/(|k|*n-1), n>=0.
For the members of the family C(k,n), k=2..9, see A130564.
The family c(k,n):=binomial((k+1)*n-2,n)/(k*n-1), n>=1, has the members A006013, A006632, A118971,for k=2,3,4 respectively (but the offset there is 0) and A130564 for k=5.

Harvey P. Dale, Table of n, a(n) for n = 1..806
Elżbieta Liszewska, Wojciech Młotkowski, Some relatives of the Catalan sequence, arXiv:1907.10725 [math.CO], 2019.

Cf. k=5 member A130564. A006013, A006632, A118971,

Table[Binomial[7n-2,n]/(6n-1),{n,20}] (* Harvey P. Dale, Feb 25 2013 *)

a(n) = binomial((k+1)*n-2,n)/(k*n-1), with k=6.
G.f.: inverse series of y*(1-y)^6.
a(n) = (6/7)*binomial(7*n,n)/(7*n-1). [Bruno Berselli, Jan 17 2014]
From Wolfdieter Lang, Feb 06 2020: (Start)
G.f.: (6/7)*(1 - hypergeom([-1, 1, 2, 3, 4, 5]/7, [1, 2, 3, 4, 5]/6, (7^7/6^6)*x)).
E.g.f.: (6/7)*(1 - hypergeom([-1, 1, 2, 3, 4, 5]/7, [1, 2, 3, 4, 5, 6]/6, (7^7/6^6)*x)). (End)

A251576 E.g.f.: exp(6xG(x)^5) / G(x)^5 where G(x) = 1 + x*G(x)^6 is the g.f. of A002295.

Original entry on oeis.org

1, 1, 6, 96, 2736, 115056, 6455376, 454666176, 38610711936, 3842344221696, 438721154343936, 56549927146392576, 8123473514799876096, 1287034084022760677376, 222964032114987212998656, 41930788886197036399190016, 8507629742037888427727486976, 1852490637585980898960109142016

Offset: 0

Paul D. Hanna, Dec 06 2014

nonn

			E.g.f.: A(x) = 1 + x + 6*x^2/2! + 96*x^3/3! + 2736*x^4/4! + 115056*x^5/5! +...
such that A(x) = exp(6*x*G(x)^5) / G(x)^5
where G(x) = 1 + x*G(x)^6 is the g.f. of A002295:
G(x) = 1 + x + 6*x^2 + 51*x^3 + 506*x^4 + 5481*x^5 + 62832*x^6 +...
Note that
A'(x) = exp(6*x*G(x)^5) = 1 + 6*x + 96*x^2/2! + 2736*x^3/3! +...
LOGARITHMIC DERIVATIVE.
The logarithm of the e.g.f. begins:
log(A(x)) = x + 5*x^2/2 + 40*x^3/3 + 385*x^4/4 + 4095*x^5/5 + 46376*x^6/6 +...
and so A'(x)/A(x) = G(x)^5.
TABLE OF POWERS OF E.G.F.
Form a table of coefficients of x^k/k! in A(x)^n as follows.
n=1: [1, 1,   6,   96,  2736,  115056,   6455376,  454666176, ...];
n=2: [1, 2,  14,  228,  6456,  268992,  14968224, 1047087648, ...];
n=3: [1, 3,  24,  402, 11376,  470808,  26011584, 1808151552, ...];
n=4: [1, 4,  36,  624, 17736,  730944,  40143456, 2774490624, ...];
n=5: [1, 5,  50,  900, 25800, 1061400,  58017600, 3989340000, ...];
n=6: [1, 6,  66, 1236, 35856, 1475856,  80395056, 5503484736, ...];
n=7: [1, 7,  84, 1638, 48216, 1989792, 108156384, 7376303088, ...];
n=8: [1, 8, 104, 2112, 63216, 2620608, 142314624, 9676910592, ...]; ...
in which the main diagonal begins (see A251586):
[1, 2, 24, 624, 25800, 1475856, 108156384, 9676910592, ...]
and is given by the formula:
[x^n/n!] A(x)^(n+1) = 6^(n-4) * (n+1)^(n-5) * (125*n^4 + 810*n^3 + 2095*n^2 + 2586*n + 1296) for n>=0.

Cf. A251586, A251666, A130564, A002295.

Cf. Variants: A243953, A251573, A251574, A251575, A251577, A251578, A251579, A251580.

Flatten[{1,1,Table[Sum[6^k * n!/k! * Binomial[6*n-k-6, n-k] * (k-1)/(n-1),{k,0,n}],{n,2,20}]}] (* Vaclav Kotesovec, Dec 07 2014 *)

{a(n) = local(G=1);for(i=1,n,G=1+x*G^6 +x*O(x^n)); n!*polcoeff(exp(6*x*G^5)/G^5, n)}
for(n=0, 20, print1(a(n), ", "))

{a(n) = if(n==0||n==1, 1, sum(k=0, n, 6^k * n!/k! * binomial(6*n-k-6,n-k) * (k-1)/(n-1) ))}
for(n=0, 20, print1(a(n), ", "))

Let G(x) = 1 + x*G(x)^6 be the g.f. of A002295, then the e.g.f. A(x) of this sequence satisfies:
(1) A'(x)/A(x) = G(x)^5.
(2) A'(x) = exp(6*x*G(x)^5).
(3) A(x) = exp( Integral G(x)^5 dx ).
(4) A(x) = exp( Sum_{n>=1} A130564(n)*x^n/n ), where A130564(n) = binomial(6*n-2,n)/(5*n-1).
(5) A(x) = F(x/A(x)) where F(x) is the e.g.f. of A251586.
(6) A(x) = Sum_{n>=0} A251586(n)*(x/A(x))^n/n! and
(7) [x^n/n!] A(x)^(n+1) = (n+1)*A251586(n),
where A251586(n) = 6^(n-4) * (n+1)^(n-6) * (125*n^4 + 810*n^3 + 2095*n^2 + 2586*n + 1296).
a(n) = Sum_{k=0..n} 6^k * n!/k! * binomial(6*n-k-6, n-k) * (k-1)/(n-1) for n>1.
Recurrence: 5*(5*n-9)*(5*n-8)*(5*n-7)*(5*n-6)*(9*n^4 - 99*n^3 + 413*n^2 - 777*n + 559)*a(n) = 72*(5832*n^9 - 113724*n^8 + 986580*n^7 - 5003586*n^6 + 16373448*n^5 - 35916483*n^4 + 52931854*n^3 - 50678109*n^2 + 28701206*n - 7357350)*a(n-1) + 46656*(9*n^4 - 63*n^3 + 170*n^2 - 212*n + 105)*a(n-2). - Vaclav Kotesovec, Dec 07 2014
a(n) ~ 6^(6*(n-1)-1/2) / 5^(5*(n-1)-1/2) * n^(n-2) / exp(n-1). - Vaclav Kotesovec, Dec 07 2014

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

cp's OEIS Frontend

A062994 Eighth column of triangle A062993 (without leading zeros). A Pfaff-Fuss or 9-Raney sequence.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A230388 a(n) = binomial(11*n+1,n)/(11*n+1).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula

A234466 a(n) = 7*binomial(8*n+7,n)/(8*n+7).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

A234513 8*binomial(9*n+8,n)/(9*n+8).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A230388 a(n) = binomial(11n+1,n)/(11n+1).

A234466 a(n) = 7binomial(8n+7,n)/(8*n+7).

A234513 8binomial(9n+8,n)/(9*n+8).

A234573 a(n) = 9binomial(10n+9,n)/(10*n+9).

A251576 E.g.f.: exp(6xG(x)^5) / G(x)^5 where G(x) = 1 + x*G(x)^6 is the g.f. of A002295.

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