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.

Showing 1-9 of 9 results.

A051945 a(n) = C(n)*(5*n+1) where C(n) = Catalan numbers (A000108).

Original entry on oeis.org

1, 6, 22, 80, 294, 1092, 4092, 15444, 58630, 223652, 856596, 3292016, 12688732, 49031400, 189885240, 736808220, 2863971270, 11149451940, 43465121700, 169657266240, 662976162420, 2593424304120, 10154564564040, 39794915183400, 156078401826204, 612605246582952
Offset: 0

Views

Author

Barry E. Williams, Dec 20 1999

Keywords

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

Crossrefs

Column k=5 of A330965.
Cf. A016813, A000108, A051924.

Programs

  • Magma
    [Catalan(n)*(5*n+1):n in [0..27] ]; // Marius A. Burtea, Jan 05 2020
  • Magma
    R:=PowerSeriesRing(Rationals(),29); (Coefficients(R!((2-3*x-2*Sqrt(1-4*x))/(x*Sqrt(1-4*x))))); // Marius A. Burtea, Jan 05 2020
  • Mathematica
    Table[CatalanNumber[n](5n+1),{n,0,30}] (* Harvey P. Dale, Jul 27 2020 *)
  • PARI
    a(n) = (5*n+1)*binomial(2*n, n)/(n+1)  \\ Michel Marcus, Jul 12 2013

Formula

(n+1)*(5n-4)*a(n) - 2*(5n+1)(2n-1)*a(n-1) = 0. - R. J. Mathar, Jul 09 2012
G.f.: (2 - 3*x - 2*sqrt(1 - 4*x))/(x*sqrt(1 - 4*x)). - Ilya Gutkovskiy, Jun 13 2017
From Peter Bala, Aug 23 2025: (Start)
a(n) = binomial(2*n, n) + 4*binomial(2*n, n-1) = A000984(n) + 4*A001791(n).
a(n) ~ 4^n * 5/sqrt(Pi*n). (End)
E.g.f.: exp(2*x)*((1 + 5*x)*BesselI(0, 2*x) - BesselI(1, 2*x) - 5*x*BesselI(2, 2*x)). - Stefano Spezia, Aug 29 2025

Extensions

Terms a(21) and beyond from Andrew Howroyd, Jan 04 2020

A051944 a(n) = C(n)*(4*n+1) where C(n) = Catalan numbers (A000108).

Original entry on oeis.org

1, 5, 18, 65, 238, 882, 3300, 12441, 47190, 179894, 688636, 2645370, 10192588, 39373700, 152443080, 591385545, 2298248550, 8945490510, 34867625100, 136079265630, 531693754020, 2079632696700, 8141948163960, 31904544069450, 125120702290428, 491056586546652
Offset: 0

Views

Author

Barry E. Williams, Dec 20 1999

Keywords

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

Crossrefs

Column k=4 of A330965.
Cf. A016777, A000108, A051924.

Programs

  • Magma
    [Catalan(n)*(4*n+1):n in [0..30] ]; // Marius A. Burtea, Jan 05 2020
  • Magma
    R:=PowerSeriesRing(Rationals(),30); (Coefficients(R!( (3 - 4*x - 3*Sqrt(1 - 4*x))/(2*x*Sqrt(1 - 4*x)))) ); // Marius A. Burtea, Jan 05 2020
  • Mathematica
    Table[CatalanNumber[n](4n+1),{n,0,30}] (* Harvey P. Dale, Feb 21 2022 *)
  • PARI
    {a(n)=if(n<0, 0, (4*n+1)*binomial(2*n,n)/(n+1))} /* Michael Somos, Sep 17 2006 */

Formula

The Hankel determinant transform is A025172(n-1). - Michael Somos, Sep 17 2006
-(n+1)*(4*n-3)*a(n) + 2*(4*n+1)*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Nov 19 2014
G.f.: (3 - 4*x - 3*sqrt(1 - 4*x))/(2*x*sqrt(1 - 4*x)). - Ilya Gutkovskiy, Jun 13 2017
From Peter Bala, Aug 23 2025: (Start)
a(n) = binomial(2*n, n) + 3*binomial(2*n, n-1) = A000984(n) + 3*A001791(n).
a(n) ~ 4^(n+1)/sqrt(Pi*n). (End)

Extensions

Terms a(21) and beyond from Andrew Howroyd, Jan 02 2020

A050476 a(n) = C(n)*(6*n + 1) where C(n) = Catalan numbers (A000108).

Original entry on oeis.org

1, 7, 26, 95, 350, 1302, 4884, 18447, 70070, 267410, 1024556, 3938662, 15184876, 58689100, 227327400, 882230895, 3429693990, 13353413370, 52062618300, 203235266850, 794258570820, 3107215911540, 12167180964120, 47685286297350, 187036101361980, 734153906619252, 2883674432327864, 11333968799308652
Offset: 0

Views

Author

Barry E. Williams, Dec 24 1999

Keywords

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

Crossrefs

Column k=6 of A330965.
Cf. A016861, A000108, A051945.

Programs

  • Magma
    [Catalan(n)*(6*n+1):n in [0..27] ]; // Marius A. Burtea, Jan 05 2020
  • Magma
    R:=PowerSeriesRing(Rationals(),30); (Coefficients(R!( (5-8*x-5*Sqrt(1-4*x))/(2*x*Sqrt(1-4*x))))); // Marius A. Burtea, Jan 05 2020
  • Mathematica
    Table[CatalanNumber[n](6n+1),{n,0,20}] (* Harvey P. Dale, Nov 05 2011 *)

Formula

5*(n+1)*a(n) + 2*(-14*n-1)*a(n-1) + 16*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Feb 04 2015
G.f.: (5 - 8*x - 5*sqrt(1 - 4*x))/(2*x*sqrt(1 - 4*x)). - Ilya Gutkovskiy, Jun 13 2017
From Peter Bala, Aug 23 2025: (Start)
a(n) = binomial(2*n, n) + 5*binomial(2*n, n-1) = A000984(n) + 5*A001791(n).
a(n) ~ 4^n * 6/sqrt(Pi*n). (End)

A050489 a(n) = C(n)*(10*n + 1) where C(n) = Catalan numbers (A000108).

Original entry on oeis.org

1, 11, 42, 155, 574, 2142, 8052, 30459, 115830, 442442, 1696396, 6525246, 25169452, 97319900, 377096040, 1463921595, 5692584870, 22169259090, 86452604700, 337547269290, 1319388204420, 5162382341220, 20217646564440, 79246770753150, 310866899505084
Offset: 0

Views

Author

Barry E. Williams, Dec 27 1999

Keywords

References

  • Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

Crossrefs

Column k=10 of A330965.
Cf. A017173, A027810, A000108.

Programs

Formula

-(n+1)*(10*n-9)*a(n) + 2*(10*n+1)*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Dec 03 2014
From Stefano Spezia, Feb 16 2020: (Start)
O.g.f.: 2*(1 + sqrt(1 - 4*x) + 16*x)/((1 + sqrt(1 - 4*x))^2*sqrt(1 - 4*x)).
E.g.f.: exp(2*x)*(I_0(2*x) + 9*I_1(2*x)), where I_n(x) is the modified Bessel function of the first kind.
(End)
G.f.: (9 - 16*x - 9*sqrt(1 - 4*x))/(2*x*sqrt(1 - 4*x)). - Amiram Eldar, Jul 08 2023
From Peter Bala, Aug 23 2025: (Start)
a(n) = binomial(2*n, n) + 9*binomial(2*n, n-1) = A000984(n) + 9*A001791(n).
a(n) ~ 4^n * 10/sqrt(Pi*n). (End)

Extensions

Corrected and extended by Harvey P. Dale, Jul 19 2011

A050477 a(n) = C(n)*(7*n + 1) where C(n) = Catalan numbers (A000108).

Original entry on oeis.org

1, 8, 30, 110, 406, 1512, 5676, 21450, 81510, 311168, 1192516, 4585308, 17681020, 68346800, 264769560, 1027653570, 3995416710, 15557374800, 60660114900, 236813267460, 925540979220, 3621007518960, 14179797364200, 55575657411300, 217993800897756, 855702566655552
Offset: 0

Views

Author

Barry E. Williams, Dec 24 1999

Keywords

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

Crossrefs

Column k=7 of A330965.
Cf. A016921, A000108, A051945.

Programs

  • Magma
    [Catalan(n)*(7*n+1):n in [0..25] ]; // Marius A. Burtea, Jan 05 2020
  • Magma
    R:=PowerSeriesRing(Rationals(),27); (Coefficients(R!( (3-5*x-3*Sqrt(1-4*x))/(x*Sqrt(1 - 4*x))) )); // Marius A. Burtea, Jan 05 2020
  • Mathematica
    Table[CatalanNumber[n](7n+1),{n,0,30}] (* Harvey P. Dale, Jun 01 2024 *)
  • PARI
    a(n) = (7*n+1) * binomial(2*n,n)/(n+1) \\ Michel Marcus, Jul 24 2013

Formula

3*(n+1)*a(n) + (-17*n-1)*a(n-1) + 10*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Feb 13 2015
-(n+1)*(7*n-6)*a(n) + 2*(7*n+1)*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Feb 13 2015
G.f.: (3 - 5*x - 3*sqrt(1 - 4*x))/(x*sqrt(1 - 4*x)). - Ilya Gutkovskiy, Jun 13 2017
From Peter Bala, Aug 23 2025: (Start)
a(n) = binomial(2*n, n) + 6*binomial(2*n, n-1) = A000984(n) + 6*A001791(n).
a(n) ~ 4^n * 7/sqrt(Pi*n). (End)

Extensions

Terms a(21) and beyond from Andrew Howroyd, Jan 02 2020

A050478 a(n) = C(n)*(8*n+1) where C(n) = Catalan numbers (A000108).

Original entry on oeis.org

1, 9, 34, 125, 462, 1722, 6468, 24453, 92950, 354926, 1360476, 5231954, 20177164, 78004500, 302211720, 1173076245, 4561139430, 17761336230, 69257611500, 270391268070, 1056823387620
Offset: 0

Views

Author

Barry E. Williams, Dec 24 1999

Keywords

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

Crossrefs

Column k=8 of A330965.
Cf. A016993, A000108, A051945.

Programs

  • Magma
    [Catalan(n)*(8*n+1):n in [0..30]]; // Vincenzo Librandi, Jan 27 2013
  • Magma
    R:=PowerSeriesRing(Rationals(),30); (Coefficients(R!( (7-12*x-7*Sqrt(1-4*x))/(2*x*Sqrt(1-4*x))))); // Marius A. Burtea, Jan 05 2020
  • Mathematica
    Table[CatalanNumber[n](8n+1),{n,0,20}] (* Harvey P. Dale, May 20 2012 *)

Formula

-(n+1)*(8*n-7)*a(n) + 2*(8*n+1)*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Dec 03 2014
G.f.: (7 - 12*x - 7*sqrt(1 - 4*x))/(2*x*sqrt(1 - 4*x)). - Ilya Gutkovskiy, Jun 13 2017
From Peter Bala, Aug 23 2025: (Start)
a(n) = binomial(2*n, n) + 7*binomial(2*n, n-1) = A000984(n) + 7*A001791(n).
a(n) ~ 2^(2*n+3)/sqrt(Pi*n). (End)

A050479 a(n) = C(n)*(9*n + 1) where C(n) = Catalan numbers (A000108).

Original entry on oeis.org

1, 10, 38, 140, 518, 1932, 7260, 27456, 104390, 398684, 1528436, 5878600, 22673308, 87662200, 339653880, 1318498920, 5126862150, 19965297660, 77855108100, 303969268680, 1188105796020, 4648590733800, 18205030164360, 71356399639200, 279909199969308, 1098799886728152
Offset: 0

Views

Author

Barry E. Williams, Dec 24 1999

Keywords

References

  • A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

Crossrefs

Column k=9 of A330965.
Cf. A017077, A000108, A051945.

Programs

  • Magma
    [Catalan(n)*(9*n+1):n in [0..27] ]; // Marius A. Burtea, Jan 05 2020
  • Magma
    R:=PowerSeriesRing(Rationals(),30); (Coefficients(R!( (4-7*x-4*Sqrt(1-4*x))/(x*Sqrt(1-4*x))))); // Marius A. Burtea, Jan 05 2020
  • Mathematica
    A050479[n_] := CatalanNumber[n]*(9*n + 1);
Array[A050479, 30, 0] (* Paolo Xausa, Aug 24 2025 *)

Formula

4*(n+1)*a(n) + (-23*n-1)*a(n-1) + 14*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Feb 13 2015
-(n+1)*(9*n-8)*a(n) + 2*(9*n+1)*(2*n-1)*a(n-1) = 0. - R. J. Mathar, Feb 13 2015
G.f.: (4 - 7*x - 4*sqrt(1 - 4*x))/(x*sqrt(1 - 4*x)). - Ilya Gutkovskiy, Jun 13 2017
From Peter Bala, Aug 23 2025: (Start)
a(n) = binomial(2*n, n) + 8*binomial(2*n, n-1) = A000984(n) + 8*A001791(n).
a(n) ~ 4^n * 9/sqrt(Pi*n). (End)

Extensions

Terms a(21) and beyond from Andrew Howroyd, Jan 05 2020

A050490 a(n) = C(n)*(11n+1) where C(n) = Catalan numbers (A000108).

Original entry on oeis.org

1, 12, 46, 170, 630, 2352, 8844, 33462, 127270, 486200, 1864356, 7171892, 27665596, 106977600, 414538200, 1609344270, 6258307590, 24373220520, 95050101300, 371125269900, 1450670612820, 5676173948640, 22230262964520, 87137141867100, 341824599040860, 1341897206800752
Offset: 0

Views

Author

Barry E. Williams, Dec 27 1999

Keywords

References

  • Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

Crossrefs

Column k=11 of A330965.
Cf. A017281, A027810, A000108.

Programs

  • Magma
    [Catalan(n)*(11*n+1):n in [0..25] ]; // Marius A. Burtea, Jan 05 2020
  • Mathematica
    Table[CatalanNumber[n](11n+1),{n,0,20}] (* Harvey P. Dale, Jul 12 2018 *)

Formula

From R. J. Mathar, Feb 13 2015: (Start)
5*(n+1)*a(n) + (-29*n-1)*a(n-1) + 18*(2*n-3)*a(n-2) = 0.
-(n+1)*(11*n-10)*a(n) + 2*(11*n+1)*(2*n-1)*a(n-1) = 0. (End)
G.f.: (5 - 9*x - 5*sqrt(1 - 4*x))/(x*sqrt(1 - 4*x)). - Amiram Eldar, Jul 08 2023

Extensions

Terms a(21) and beyond from Andrew Howroyd, Jan 05 2020

A050491 a(n) = C(n)*(12n+1) where C(n) = Catalan numbers (A000108).

Original entry on oeis.org

1, 13, 50, 185, 686, 2562, 9636, 36465, 138710, 529958, 2032316, 7818538, 30161740, 116635300, 451980360, 1754766945, 6824030310, 26577181950, 103647597900, 404703270510, 1581953021220, 6189965556060, 24242879364600, 95027512981050, 372782298576636, 1463445866837052
Offset: 0

Views

Author

Barry E. Williams, Dec 27 1999

Keywords

References

  • Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

Links

Crossrefs

Column k=12 of A330965.
Cf. A017401, A027810, A000108.

Programs

  • Magma
    [Catalan(n)*(12*n+1):n in [0..25] ]; // Marius A. Burtea, Jan 05 2020
  • Mathematica
    Table[CatalanNumber[n] * (12*n + 1), {n, 0, 25}] (* Amiram Eldar, Jul 08 2023 *)

Formula

G.f.: (11 - 20*x - 11*sqrt(1 - 4*x))/(2*x*sqrt(1 - 4*x)). - Amiram Eldar, Jul 08 2023

Extensions

Terms a(21) and beyond from Andrew Howroyd, Jan 05 2020
Showing 1-9 of 9 results.

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