A337292 -id:A337292 - cp's OEIS frontend

A337291 a(n) = 3binomial(4n,n)/(4*n-1).

4, 12, 60, 364, 2448, 17556, 131560, 1017900, 8069424, 65204656, 535070172, 4446927732, 37353738800, 316621743480, 2704784196240, 23263187479980, 201275443944432, 1750651680235920, 15298438066553776, 134252511729576240, 1182622941581590080

Offset: 1

Lucas A. Brown, Aug 21 2020

a(n) is the number of lattice paths from (0,0) to (3n,n) using only the steps (1,0) and (0,1) and whose only lattice points on the line y = x/3 are the path's endpoints. - Lucas A. Brown, Aug 21 2020

R. J. Mathar, The Eggenberger-Polya urn process: Probabilities of revisited ball ratios, vixra:2502 (2025)

Cf. A006632, A337292.

Array[3 Binomial[4 #, #]/(4 # - 1) &, 21] (* Michael De Vlieger, Aug 21 2020 *)

a(n) = {3*binomial(4*n,n)/(4*n-1)} \\ Andrew Howroyd, Aug 21 2020

a(n) = 4*A006632(n).
G.f.: 4*x*F(x)^3 where F(x) = 1 + x*F(x)^4 is the g.f. of A002293.
D-finite with recurrence 3*n*(3*n-1)*(3*n-2)*a(n) -8*(4*n-5)*(4*n-3)*(2*n-1)*a(n-1)=0, a(0)=1. - R. J. Mathar, Jan 26 2025

A337350 a(n) is the number of lattice paths from (0,0) to (2n,2n) using only the steps (1,0) and (0,1) and which do not touch any other points of the form (2k,2k).

Original entry on oeis.org

1, 6, 34, 300, 3146, 36244, 443156, 5646040, 74137050, 996217860, 13633173180, 189347631720, 2662142601924, 37815138677960, 541882155414376, 7823955368697776, 113712609033955834, 1662288563798703204, 24424940365489658540, 360537080085493670856

Offset: 0

Lucas A. Brown, Aug 24 2020

The terms of this sequence may be computed via a determinant; see Lemma 10.7.2 of the Krattenthaler reference for details.

Christian Krattenthaler, Lattice path enumeration, In: Handbook of Enumerative Combinatorics. Edited by Miklos Bona. CRC Press, 2015, pages 589-678.
R. J. Mathar, The Eggenberger-Polya urn process: Probabilities of revisited ball ratios, vixra:2502.0097 (2025) Table 4
Quy Nhan, A finite sum of binomial coefficients, MathStackExchange, 2025-06-21.

Cf. A000108, A001448, A337291, A337292, A337351, A337352.

seq(n)={Vec(2 - 1/(O(x*x^n) + sum(k=0, n, binomial(4*k,2*k)*x^k)))} \\ Andrew Howroyd, Aug 25 2020

G.f.: 2 - 1 / (Sum_{n>=0} binomial(4*n,2*n) * x^n).
a(n) = binomial(4*n,2*n) * (8*n+1) / (8*n^2 + 2*n - 1) for n >= 1.  For proof, see the Quy Nhan link.
D-finite with recurrence n*(2*n+1)*(8*n-7)*a(n) -2*(4*n-5)*(4*n-3)*(8*n+1)*a(n-1)=0. - R. J. Mathar, Jan 26 2025
From Lucas A. Brown, Jul 13 2025: (Start)
G.f.: 2 - sqrt(2-32*x) / sqrt(1+sqrt(1-16*x)).
a(n) = A000108(2*n) + 4 * A000108(2*n-1). (End)

A337351 a(n) is the number of lattice paths from (0,0) to (3n,2n) using only the steps (1,0) and (0,1) and which do not touch any other points of the form (3k,2k).

Original entry on oeis.org

1, 10, 110, 1805, 34770, 731760, 16295600, 377438250, 8999246900, 219399101415, 5444124108810, 137040309706725, 3490834454580950, 89816746611096280, 2330761164942308080, 60932036847971297230, 1603218808449019802550, 42423276620326253035205

Offset: 0

Lucas A. Brown, Aug 24 2020

The terms of this sequence may be computed via a determinant; see Lemma 10.7.2 of the Krattenthaler reference for details.

Christian Krattenthaler, "Lattice path enumeration". In: Handbook of Enumerative Combinatorics. Edited by Miklos Bona. CRC Press, 2015, pages 589-678.
R. J. Mathar, The Eggenberger-Polya urn process: Probabilities of revisited ball ratios, vixra:2502.0097 (2025) Table 4

Cf. A337291, A337292, A337350, A337352.

seq(n)={Vec(2 - 1/(O(x*x^n) + sum(k=0, n, binomial(5*k,2*k)*x^k)))} \\ Andrew Howroyd, Aug 25 2020

G.f.: 2 - 1 / (Sum_{n>=0} binomial(5*n,2*n) * x^n).

A337352 a(n) is the number of lattice paths from (0,0) to (3n,3n) using only the steps (1,0) and (0,1) and which do not touch any other points of the form (3k,3k).

Original entry on oeis.org

1, 20, 524, 19660, 854380, 40304080, 2004409236, 103440770760, 5486614131756, 297239307415792, 16376472734974384, 914734188877259884, 51680064605716043636, 2948046519564292501232, 169560941932509940657016, 9822377923336683964009296, 572554753384166308597716396

Offset: 0

Lucas A. Brown, Aug 24 2020

The terms of this sequence may be computed via a determinant; see Lemma 10.7.2 of the Krattenthaler reference for details.

Christian Krattenthaler, "Lattice path enumeration". In: Handbook of Enumerative Combinatorics. Edited by Miklos Bona. CRC Press, 2015, pages 589-678.
R. J. Mathar, The Eggenberger-Polya urn process: Probabilities of revisited ball ratios, vixra:2502.0097 (2025) Table 4

Cf. A337291, A337292, A337350, A337351.

seq(n)={Vec(2 - 1/(O(x*x^n) + sum(k=0, n, binomial(6*k,3*k)*x^k)))} \\ Andrew Howroyd, Aug 25 2020

G.f.: 2 - 1 / (Sum_{n>=0} binomial(6*n,3*n) * x^n).

cp's OEIS Frontend

A337291 a(n) = 3*binomial(4*n,n)/(4*n-1).

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A337350 a(n) is the number of lattice paths from (0,0) to (2n,2n) using only the steps (1,0) and (0,1) and which do not touch any other points of the form (2k,2k).

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Author

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Programs

PARI

Formula

A337351 a(n) is the number of lattice paths from (0,0) to (3n,2n) using only the steps (1,0) and (0,1) and which do not touch any other points of the form (3k,2k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A337352 a(n) is the number of lattice paths from (0,0) to (3n,3n) using only the steps (1,0) and (0,1) and which do not touch any other points of the form (3k,3k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

A337291 a(n) = 3binomial(4n,n)/(4*n-1).