A051945 -id:A051945 - cp's OEIS frontend

A330965 Array read by descending antidiagonals: A(n,k) = (1 + kn)C(n) where C(n) = Catalan numbers (A000108).

1, 1, 1, 1, 2, 2, 1, 3, 6, 5, 1, 4, 10, 20, 14, 1, 5, 14, 35, 70, 42, 1, 6, 18, 50, 126, 252, 132, 1, 7, 22, 65, 182, 462, 924, 429, 1, 8, 26, 80, 238, 672, 1716, 3432, 1430, 1, 9, 30, 95, 294, 882, 2508, 6435, 12870, 4862, 1, 10, 34, 110, 350, 1092, 3300, 9438, 24310, 48620, 16796

Offset: 0

Andrew Howroyd, Jan 04 2020

			Array begins:
====================================================
n\k |   0    1    2    3     4     5     6     7
----+-----------------------------------------------
  0 |   1    1    1    1     1     1     1     1 ...
  1 |   1    2    3    4     5     6     7     8 ...
  2 |   2    6   10   14    18    22    26    30 ...
  3 |   5   20   35   50    65    80    95   110 ...
  4 |  14   70  126  182   238   294   350   406 ...
  5 |  42  252  462  672   882  1092  1302  1512 ...
  6 | 132  924 1716 2508  3300  4092  4884  5676 ...
  7 | 429 3432 6435 9438 12441 15444 18447 21450 ...
  ...

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

Andrew Howroyd, Table of n, a(n) for n = 0..1325

Columns k=0..12 are A000108, A000984, A001700, A051924, A051944, A051945, A050476, A050477, A050478, A050479, A050489, A050490, A050491.

A330965[n_, k_] := CatalanNumber[n]*(k*n + 1);
Table[A330965[k, n - k], {n, 0, 10}, {k, 0, n}] (* Paolo Xausa, Aug 24 2025 *)

T(n, k)={(1 + k*n)*binomial(2*n,n)/(n+1)}

A(n,k) = (1 + k*n)*binomial(2*n,n)/(n+1).
A(n,k) = 2*(k*n+1)*(2*n-1)*A(n-1,k)/((n+1)*(k*n-k+1)) for n > 0.
G.f. of column k: (k - 1 - (2*k-4)*x - (k-1)*sqrt(1 - 4*x))/(2*x*sqrt(1 - 4*x)).

A050476 a(n) = C(n)(6n + 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

Barry E. Williams, Dec 24 1999

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

Andrew Howroyd, Table of n, a(n) for n = 0..200

Column k=6 of A330965.

Cf. A016861, A000108, A051945.

[Catalan(n)*(6*n+1):n in [0..27] ]; // Marius A. Burtea, Jan 05 2020

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

Table[CatalanNumber[n](6n+1),{n,0,20}] (* Harvey P. Dale, Nov 05 2011 *)

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)

A050477 a(n) = C(n)(7n + 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

Barry E. Williams, Dec 24 1999

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

Andrew Howroyd, Table of n, a(n) for n = 0..200

Column k=7 of A330965.

Cf. A016921, A000108, A051945.

[Catalan(n)*(7*n+1):n in [0..25] ]; // Marius A. Burtea, Jan 05 2020

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

Table[CatalanNumber[n](7n+1),{n,0,30}] (* Harvey P. Dale, Jun 01 2024 *)

a(n) = (7*n+1) * binomial(2*n,n)/(n+1) \\ Michel Marcus, Jul 24 2013

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)

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

A050478 a(n) = C(n)(8n+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

Barry E. Williams, Dec 24 1999

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

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Column k=8 of A330965.

Cf. A016993, A000108, A051945.

[Catalan(n)*(8*n+1):n in [0..30]]; // Vincenzo Librandi, Jan 27 2013

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

Table[CatalanNumber[n](8n+1),{n,0,20}] (* Harvey P. Dale, May 20 2012 *)

-(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)(9n + 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

Barry E. Williams, Dec 24 1999

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

Andrew Howroyd, Table of n, a(n) for n = 0..200

Column k=9 of A330965.

Cf. A017077, A000108, A051945.

[Catalan(n)*(9*n+1):n in [0..27] ]; // Marius A. Burtea, Jan 05 2020

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

A050479[n_] := CatalanNumber[n]*(9*n + 1);
Array[A050479, 30, 0] (* Paolo Xausa, Aug 24 2025 *)

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)

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

A052204 a(n) = (5n+1)*C(4n,n)/(3n+1).

Original entry on oeis.org

1, 6, 44, 352, 2940, 25194, 219604, 1937520, 17250012, 154663960, 1394538288, 12631852688, 114858935204, 1047772373340, 9584557428600, 87885886492320, 807564936805020, 7434289153896264, 68551275793965328, 633038816547052800

Offset: 0

Barry E. Williams, Jan 28 2000

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A002293, A051945.

[(5*n+1)*Binomial(4*n,n)/(3*n+1) : n in [0..20]]; // Wesley Ivan Hurt, Aug 10 2016

A052204:=n->(5*n+1)*binomial(4*n,n)/(3*n+1): seq(A052204(n), n=0..20); # Wesley Ivan Hurt, Aug 10 2016

Table[(5 n + 1) Binomial[4 n, n]/(3 n + 1), {n, 0, 20}] (* Wesley Ivan Hurt, Aug 10 2016 *)

for(n=0,25, print1((5*n+1)*binomial(4*n,n)/(3*n+1), ", ")) \\ G. C. Greubel, Feb 16 2017

G.f.: (2*g-1)*g/(4-3*g) where g = 1+x*g^4 is the g.f. of A002293. - Mark van Hoeij, Nov 11 2011
Conjecture: 6*n*(3*n-1)*(3*n+1)*a(n) + (-809*n^3 + 1444*n^2 - 1505*n + 582)*a(n-1) + 88*(4*n-5)*(4*n-7)*(2*n-3)*a(n-2) = 0. - R. J. Mathar, Sep 29 2012
a(n) ~ 5*2^(8*n+1/2)*3^(-3*n-3/2)/sqrt(Pi*n). - Ilya Gutkovskiy, Aug 10 2016

More terms from James Sellers, Jan 31 2000

cp's OEIS Frontend

A330965 Array read by descending antidiagonals: A(n,k) = (1 + k*n)*C(n) where C(n) = Catalan numbers (A000108).

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

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

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

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Magma

Magma

Mathematica

Formula

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

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Magma

Mathematica

Formula

Extensions

A052204 a(n) = (5n+1)*C(4n,n)/(3n+1).

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Magma

Maple

Mathematica

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Formula

Extensions

A330965 Array read by descending antidiagonals: A(n,k) = (1 + kn)C(n) where C(n) = Catalan numbers (A000108).

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

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

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

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