A234510 -id:A234510

A232265

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

+10
14

1, 10, 135, 2100, 35475, 632502, 11714745, 223198440, 4346520750, 86128357150, 1731030945644, 35202562937100, 723029038312230, 14976976398326250, 312522428615310000, 6563314391270476752, 138617681440915119975, 2942332729799060033100, 62735156704285184848950

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Fuss-Catalan sequence is a(n,p,r) = r*binomial(n*p + r,n)/(n*p + r), where p = 9, r = 10.

LINKS

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

FORMULA

G.f. satisfies: A(x) = {1 + x*A(x)^(p/r)}^r, where p = 9, r = 10.

From _Peter Bala, Oct 16 2015: (Start)

O.g.f. A(x) = 1/x * series reversion (x*C(-x)^10), 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)^(1/10) is the o.g.f. for A062994. (End)

D-finite with recurrence: 128*n*(8*n+3)*(4*n+3)*(8*n+9)*(2*n+1)*(8*n+7)*(4*n+5)*(8*n+5)*a(n) -81*(9*n+2)*(9*n+4)*(3*n+2)*(9*n+8)*(9*n+1)*(3*n+1)*(9*n+5)*(9*n+7)*a(n-1)=0. - R. J. Mathar, Feb 21 2020

MATHEMATICA

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

PROG

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

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

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

CROSSREFS

Cf. A000108, A143554, A234505, A234506, A234507, A234508, A234509, A234510, A234513.

Cf. A062994, A000245 (k = 3), A006629 (k = 4), A196678 (k = 5), A233668 (k = 6), A233743 (k = 7), A233835 (k = 8), A234467 (k = 9), A229963 (k = 11).

KEYWORD

nonn,easy

AUTHOR

Tim Fulford, Dec 28 2013

STATUS

approved

A234465

a(n) = 3*binomial(8*n+6,n)/(4*n+3).

+10
14

1, 6, 63, 812, 11655, 178794, 2869685, 47593176, 809172936, 14028048650, 247039158366, 4406956913268, 79470057050020, 1446283758823470, 26529603944225670, 489989612605050800, 9104498753815680600, 170073237411754811568, 3192081704235788729043

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p = 8, r = 6.

LINKS

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

FORMULA

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p = 8, r = 6.

O.g.f. A(x) = 1/x * series reversion (x/C(x)^6), where C(x) is the o.g.f. for the Catalan numbers A000108. A(x)^(1/6) is the o.g.f. for A007556. - Peter Bala, Oct 14 2015

D-finite with recurrence: 7*n*(7*n+4)*(7*n+1)*(7*n+5)*(7*n+2)*(7*n+6)*(7*n+3)*a(n) -128*(8*n+3)*(4*n-1)*(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

MATHEMATICA

Table[3 Binomial[8 n + 6, n]/(4 n + 3), {n, 0, 40}] (* Vincenzo Librandi, Dec 26 2013 *)

PROG

(PARI) a(n) = 3*binomial(8*n+6, n)/(4*n+3);

(PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(4/3))^6+x*O(x^n)); polcoeff(B, n)}

(Magma) [3*Binomial(8*n+6, n)/(4*n+3): n in [0..30]]; // Vincenzo Librandi, Dec 26 2013

CROSSREFS

Cf. A000108, A007556, A234461, A234462, A234463, A234464, A234466, A234467, A230390, A007556, A069271, A118970, A212073, A233834, A234510, A234571, A235339.

KEYWORD

nonn,easy

AUTHOR

Tim Fulford, Dec 26 2013

STATUS

approved

A234571

a(n) = 4*binomial(10*n+8,n)/(5*n+4).

+10
14

1, 8, 108, 1776, 32430, 632016, 12876864, 270964320, 5843355957, 128462407840, 2868356980060, 64869895026144, 1482877843096650, 34207542810153216, 795318309360948240, 18617396126132233920, 438423206616057162258, 10379232525028947311160, 246878659984195222962220

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), where p = 10, r = 8.

LINKS

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

FORMULA

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p = 10, r = 8.

O.g.f. A(x) = 1/x * series reversion (x/C(x)^8), where C(x) is the o.g.f. for the Catalan numbers A000108. A(x)^(1/8) is the o.g.f. for A059968. - Peter Bala, Oct 14 2015

MATHEMATICA

Table[4 Binomial[10 n + 8, n]/(5 n + 4), {n, 0, 30}]

PROG

(PARI) a(n) = 4*binomial(10*n+8, n)/(5*n+4);

(PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(5/4))^8+x*O(x^n)); polcoeff(B, n)}

(Magma) [4*Binomial(10*n+8, n)/(5*n+4): n in [0..30]];

CROSSREFS

Cf. A059968, A234525, A234526, A234527, A234528, A234529, A234570, A234573, A059968, A069271, A118970, A212073, A233834, A234465, A234510, A235339.

KEYWORD

nonn,easy

AUTHOR

Tim Fulford, Dec 28 2013

STATUS

approved

A212073

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

+10
12

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

LINKS

Table of n, a(n) for n=0..19.

Wikipedia, Fuss-Catalan number

FORMULA

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

EXAMPLE

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 +...

MATHEMATICA

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 *)

PROG

(PARI) {a(n)=binomial(6*n+4, n) * 4/(6*n+4)}

for(n=0, 40, print1(a(n), ", "))

(PARI) {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)}

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Paul D. Hanna, Apr 29 2012

STATUS

approved

A234513

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

+10
12

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), where p=9, r=8.

LINKS

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.

FORMULA

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)

MATHEMATICA

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

PROG

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

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

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

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Tim Fulford, Dec 27 2013

STATUS

approved

A233834

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

+10
10

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p = 7, r = 5.

LINKS

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

FORMULA

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

MATHEMATICA

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

PROG

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

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

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

CROSSREFS

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

KEYWORD

nonn,easy

AUTHOR

Tim Fulford, Dec 16 2013

STATUS

approved

A234505

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

+10
9

1, 2, 19, 252, 3885, 65274, 1159587, 21421248, 407337153, 7920326700, 156753610013, 3147328992080, 63951322669065, 1312575792628356, 27172514322677625, 566707337222428800, 11896007334177739113, 251142622845893276190, 5328891499524964282170

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), where p=9, r=2.

LINKS

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.

Sheng-liang Yang and Mei-yang Jiang, Pattern avoiding problems on the hybrid d-trees, J. Lanzhou Univ. Tech., (China, 2023) Vol. 49, No. 2, 144-150. (in Mandarin)

FORMULA

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=9, r=2.

a(n) = 2*binomial(9n+1,n-1)/n for n>0, a(0)=1. [Bruno Berselli, Jan 19 2014]

MATHEMATICA

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

PROG

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

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

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

CROSSREFS

Cf. A000108, A143554, A234506, A234507, A234508, A234509, A234510, A234513, A232265.

KEYWORD

nonn

AUTHOR

Tim Fulford, Dec 27 2013

STATUS

approved

A234506

a(n) = binomial(9*n+3, n)/(3*n+1).

+10
8

1, 3, 30, 406, 6327, 107019, 1909908, 35399520, 674842149, 13147742322, 260626484118, 5239783981320, 106585537781775, 2189670831627678, 45366284782209600, 946815917066740800, 19887218367823853937, 420076689292591271325, 8917736795123409615060, 190161017612160607167948, 4071301730663135449185705

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Fuss-Catalan sequence is a(n,p,r) = r*binomial(n*p + r, n)/(n*p + r), where p=9, r=3.

LINKS

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.

FORMULA

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=9, r=3.

MATHEMATICA

Table[Binomial[9n+3, n]/(3n+1), {n, 0, 30}]

PROG

(PARI) a(n) = binomial(9*n+3, n)/(3*n+1);

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

(Magma) [Binomial(9*n+3, n)/(3*n+1): n in [0..30]];

(Sage) [binomial(9*n+3, n)/(3*n+1) for n in (0..30)] # G. C. Greubel, Feb 09 2021

CROSSREFS

Cf. A000108, A143554, A232265, A234505, A234507, A234508, A234509, A234510, A234513.

KEYWORD

nonn

AUTHOR

Tim Fulford, Dec 27 2013

STATUS

approved

A234507

4*binomial(9*n+4,n)/(9*n+4).

+10
8

1, 4, 42, 580, 9139, 155664, 2791404, 51919296, 992414925, 19375620264, 384734333698, 7745767624560, 157746595917027, 3243956787596560, 67267249849483200, 1404952651131292800, 29529506061314207361, 624113938377564174540, 13256095235994257535900, 282803564653982441429256, 6057302574889055180495805

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), where p=9, r=4.

LINKS

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.

FORMULA

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=9, r=4.

MATHEMATICA

Table[4 Binomial[9 n + 4, n]/(9 n + 4), {n, 0, 30}]

PROG

(PARI) a(n) = 4*binomial(9*n+4, n)/(9*n+4);

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

(Magma) [1*Binomial(9*n+1, n)/(9*n+1): n in [0..30]];

CROSSREFS

Cf. A000108, A143554, A234505, A234506, A234508, A234509, A234510, A234513, A232265.

KEYWORD

nonn

AUTHOR

Tim Fulford, Dec 27 2013

STATUS

approved

A234508

5*binomial(9*n+5,n)/(9*n+5).

+10
8

1, 5, 55, 775, 12350, 211876, 3818430, 71282640, 1366368375, 26735839650, 531838637759, 10723307329700, 218658647805780, 4501362056183300, 93426735902060000, 1952884185072496992, 41074876852203972645, 868669222741822476975, 18460669540059117038250, 394033629095915025876750, 8443512680148379948569910

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), where p=9, r=5.

LINKS

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.

FORMULA

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, where p=9, r=5.

MATHEMATICA

Table[5 Binomial[9 n + 5, n]/(9 n + 5), {n, 0, 30}]

PROG

(PARI) a(n) = 5*binomial(9*n+5, n)/(9*n+5);

(PARI) {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(9/5))^5+x*O(x^n)); polcoeff(B, n)}

(Magma) [5*Binomial(9*n+5, n)/(9*n+5): n in [0..30]];

CROSSREFS

Cf. A000108, A143554, A234505, A234506, A234507, A234509, A234510, A234513, A232265.

KEYWORD

nonn

AUTHOR

Tim Fulford, Dec 27 2013

STATUS

approved