cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

User: Tim Fulford

Tim Fulford's wiki page.

Tim Fulford has authored 55 sequences. Here are the ten most recent ones:

A235340 a(n) = 10*binomial(11*n+10,n)/(11*n+10).

Original entry on oeis.org

1, 10, 155, 2870, 58565, 1270752, 28765650, 671650110, 16057800980, 391139588190, 9672348219898, 242182964452000, 6127720969229265, 156431295179478200, 4024231652469275640, 104218796026870015374, 2714941275486017847825
Offset: 0

Views

Author

Tim Fulford, Jan 06 2014

Keywords

Comments

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=11, r=10.

Links

Crossrefs

Cf. A130564, A230388, A234868, A234869, A234870, A234871, A234872, A234873, A235338, A235339.

Programs

  • Magma
    [10*Binomial(11*n+10, n)/(11*n+10): n in [0..30]];
  • Mathematica
    Table[10 Binomial[11 n + 10, n]/(11 n + 10), {n, 0, 30}]
  • PARI
    a(n) = 10*binomial(11*n+10,n)/(11*n+10);
  • PARI
    {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(11/10))^10+x*O(x^n)); polcoeff(B, n)}

Formula

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, here p=11, r=10.
From Wolfdieter Lang, Feb 15 2024: (Start)
a(n) = binomial(11*n + 9, n + 1)/(10*n + 9) which is instance k = 10 of c(k, n+1) given in a comment in A130564.
x*B(x), with the g.f. above named B(x), is the compositional inverse of y*(1 - y)^10, hence B(x)*(1 - x*B(x))^10 = 1.
G.f.: 11F10([10..20]/11, [11..20]/10; (11^11/10^10)*x) = (10/(11*x))*(1 - 10F9([-1,1,2,3,4,5,6,7,8,9]/11, [1,2,3,4,5,6,7,8,9]/10; (11^11/10^10)*x)).
(End)

A235339 a(n) = 9*binomial(11*n+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

Views

Author

Tim Fulford, Jan 06 2014

Keywords

Comments

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

Links

Crossrefs

Cf. A230388, A234868, A234869, A234870, A234871, A234872, A234873, A235338, A235340, A069271, A118970, A212073, A230388, A233834, A234465, A234510, A234571.

Programs

  • Magma
    [9*Binomial(11*n+9, n)/(11*n+9): n in [0..30]];
  • Mathematica
    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);
  • PARI
    {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(11/9))^9+x*O(x^n)); polcoeff(B, n)}

Formula

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

A235338 a(n) = 8*binomial(11*n+8,n)/(11*n+8).

Original entry on oeis.org

1, 8, 116, 2080, 41650, 892552, 20027112, 464550336, 11050084695, 268070745800, 6607118937848, 164979021222400, 4164615224071926, 106105019316578800, 2724883054841727200, 70462458864489354624, 1833143662625459289495
Offset: 0

Views

Author

Tim Fulford, Jan 06 2014

Keywords

Comments

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=11, r=8.

Links

Crossrefs

Cf. A230388, A234868, A234869, A234870, A234871, A234872, A234873, A235339, A235340.

Programs

  • Magma
    [8*Binomial(11*n+8, n)/(11*n+8): n in [0..30]];
  • Mathematica
    Table[8 Binomial[11 n + 8, n]/(11 n + 8), {n, 0, 30}]
  • PARI
    a(n) = 8*binomial(11*n+8, n)/(11*n+8);
  • PARI
    {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(11/8))^8+x*O(x^n)); polcoeff(B, n)}

Formula

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, here p=11, r=8.

A234873 7*binomial(11*n+7,n)/(11*n+7).

Original entry on oeis.org

1, 7, 98, 1729, 34300, 730597, 16323468, 377447148, 8956384437, 216859117475, 5336519142108, 133078780790725, 3355661187741408, 85414540549845934, 2191753761503128400, 56636249639625891144, 1472525237190942707955
Offset: 0

Views

Author

Tim Fulford, Jan 01 2014

Keywords

Comments

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=11, r=7.

Links

Crossrefs

Cf. A230388, A234868, A234869, A234870, A234871, A234872.

Programs

  • Magma
    [7*Binomial(11*n+7,n)/(11*n+7): n in [0..30]]; // Vincenzo Librandi, Jan 01 2014
  • Mathematica
    Table[7 Binomial[11 n + 7, n]/(11 n + 7), {n, 0, 30}] (* Vincenzo Librandi, Jan 01 2014 *)
  • PARI
    a(n) = 7*binomial(11*n+7,n)/(11*n+7);
  • PARI
    {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(11/7))^7+x*O(x^n)); polcoeff(B, n)}

Formula

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, with p=11, r=7.

A234869 3*binomial(11*n+3,n)/(11*n+3).

Original entry on oeis.org

1, 3, 36, 595, 11385, 237006, 5212064, 119126865, 2801765835, 67365151700, 1648369018296, 40914062713953, 1027625691201200, 26069631471224820, 667024542735629400, 17193066926119888716, 446028709678732029135, 11636873606948476550895
Offset: 0

Views

Author

Tim Fulford, Jan 01 2014

Keywords

Comments

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=11, r=3.

Links

Crossrefs

Cf. A230388, A234868, A234870, A234871, A234872, A234873.

Programs

  • Magma
    [3*Binomial(11*n+3,n)/(11*n+3): n in [0..30]]; // Vincenzo Librandi, Jan 01 2014
  • Mathematica
    Table[3 Binomial[11 n + 3, n]/(11 n + 3), {n, 0, 30}] (* Vincenzo Librandi, Jan 01 2014 *)
  • PARI
    a(n) = 3*binomial(11*n+3,n)/(11*n+3);
  • PARI
    {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(11/3))^3+x*O(x^n)); polcoeff(B, n)}

Formula

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, with p=11, r=3.

A234872 a(n) = 6*binomial(11*n+6,n)/(11*n+6).

Original entry on oeis.org

1, 6, 81, 1406, 27636, 585162, 13019909, 300138696, 7105216833, 171717015470, 4219267597578, 105085831400550, 2647012241261856, 67316157557021436, 1726006087183713615, 44570883175043934384, 1158139943222389790715
Offset: 0

Views

Author

Tim Fulford, Jan 01 2014

Keywords

Comments

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=11, r=6.

Links

Crossrefs

Cf. A230388, A234868, A234869, A234870, A234871, A234873.

Programs

  • Magma
    [6*Binomial(11*n+6,n)/(11*n+6): n in [0..30]]; // Vincenzo Librandi, Jan 01 2014
  • Mathematica
    Table[6 Binomial[11 n + 6, n]/(11 n + 6), {n, 0, 40}] (* Vincenzo Librandi, Jan 01 2014 *)
  • PARI
    a(n) = 6*binomial(11*n+6,n)/(11*n+6);
  • PARI
    {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(11/6))^6+x*O(x^n)); polcoeff(B, n)}

Formula

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, with p=11, r=6.
a(n) ~ 3*4^(-3-5*n)*5^(-13/2-10*n)*11^(11/2+11*n)/(n^(3/2)*sqrt(Pi)). - Stefano Spezia, Aug 23 2025

A234871 a(n) = 5*binomial(11*n+5,n)/(11*n+5).

Original entry on oeis.org

1, 5, 65, 1110, 21620, 455126, 10085845, 231814440, 5475346305, 132090011900, 3240886705386, 80621405042750, 2028732009726240, 51548408940061460, 1320738410528418175, 34083616545621832176, 885134579074202142075, 23114512490211287029665
Offset: 0

Views

Author

Tim Fulford, Jan 01 2014

Keywords

Comments

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=11, r=5.

Links

Crossrefs

Cf. A230388, A234868, A234869, A234870, A234872, A234873.

Programs

  • Magma
    [5*Binomial(11*n+5,n)/(11*n+5): n in [0..30]]; // Vincenzo Librandi, Jan 01 2014
  • Mathematica
    Table[5 Binomial[11 n + 5, n]/(11 n + 5), {n, 0, 40}] (* Vincenzo Librandi, Jan 01 2014 *)
  • PARI
    a(n) = 5*binomial(11*n+5,n)/(11*n+5);
  • PARI
    {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(11/5))^5+x*O(x^n)); polcoeff(B, n)}

Formula

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, with p=11, r=5.

A234870 4*binomial(11*n+4,n)/(11*n+4).

Original entry on oeis.org

1, 4, 50, 840, 16215, 339416, 7492342, 171714400, 4046995095, 97464474800, 2388021782916, 59337354111464, 1491714713034000, 37872300445759440, 969656048236053850, 25008097347083474496, 649098691321081570855, 16942574600154870074100
Offset: 0

Views

Author

Tim Fulford, Jan 01 2014

Keywords

Comments

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=11, r=4.

Links

Crossrefs

Cf. A230388, A234868, A234869, A234871, A234872, A234873.

Programs

  • Magma
    [4*Binomial(11*n+4,n)/(11*n+4): n in [0..30]]; // Vincenzo Librandi, Jan 01 2014
  • Mathematica
    Table[4 Binomial[11 n + 4, n]/(11 n + 4), {n, 0, 40}] (* Vincenzo Librandi, Jan 01 2014 *)
  • PARI
    a(n) = 4*binomial(11*n+4,n)/(11*n+4);
  • PARI
    {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(11/4))^4+x*O(x^n)); polcoeff(B, n)}

Formula

G.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r, with p=11, r=4.

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

Views

Author

Tim Fulford, Jan 01 2014

Keywords

Comments

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

Links

Crossrefs

Cf. A234868-A234873, A130564.

Programs

  • Magma
    [Binomial(11*n+1,n)/(11*n+1): n in [0..30]]; // Vincenzo Librandi, Jan 01 2014
  • Maple
    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
  • Mathematica
    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);
  • PARI
    {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(11))+x*O(x^n)); polcoeff(B, n)}

Formula

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

A234868 a(n) = 2*binomial(11*n+2,n)/(11*n+2).

Original entry on oeis.org

1, 2, 23, 374, 7095, 146916, 3219216, 73386170, 1722567143, 41352865400, 1010607195741, 25058477434562, 628845572227600, 15941429819185752, 407626109449551300, 10501154649486399096, 272294680440574235015, 7101160966497659412010, 186134223613500403098396
Offset: 0

Views

Author

Tim Fulford, Jan 01 2014

Keywords

Comments

Fuss-Catalan sequence is a(n,p,r) = r*binomial(np+r,n)/(np+r), this is the case p=11, r=2; also, g.f. satisfies: B(x) = {1 + x*B(x)^(p/r)}^r.

Links

Crossrefs

Cf. A230388, A234869, A234870, A234871, A234872, A234873.

Programs

  • Magma
    [2*Binomial(11*n+2,n)/(11*n+2): n in [0..30]]; // Vincenzo Librandi, Jan 01 2014
  • Mathematica
    Table[2 Binomial[11 n + 2, n]/(11 n + 2), {n, 0, 30}] (* Vincenzo Librandi, Jan 01 2014 *)
  • PARI
    a(n) = 2*binomial(11*n+2,n)/(11*n+2)
for(n=0,20, print(a(n))) \\ Sequence
  • PARI
    {a(n)=local(B=1); for(i=0, n, B=(1+x*B^(11/2))^2+x*O(x^n)); polcoeff(B, n)}
for (n=0,20, print(a(n))) \\ Generating Function
  • Sage
    [2*binomial(11*n+2,n)/(11*n+2) for n in range(20)] # F. Chapoton; Apr 29 2020

Formula

a(n) = 2*binomial(11*n+1,n-1)/n for n>0, a(0)=1. [Bruno Berselli, Jan 19 2014]

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