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).

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

Offset

0,2

Comments

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

Links

Andrew Howroyd, Table of n, a(n) for n = 0..500

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.

Formula

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) for n >= 1. (End)

a(n) ~ 2^(4*n-1/2) / (n^(3/2) * sqrt(Pi)). - Amiram Eldar, Nov 21 2025

Mathematica

a[n_] := CatalanNumber[2*n] + 4 * CatalanNumber[2*n-1]; a[0] = 1; Array[a, 20, 0] (* Amiram Eldar, Nov 21 2025 *)

Prog

(PARI) 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

Crossrefs

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

Sequence in context: A218685 A108432 A355887 * A245466 A362627 A284330

Adjacent sequences: A337347 A337348 A337349 * A337351 A337352 A337353

Keyword

nonn,easy

Author

Lucas A. Brown, Aug 24 2020

Status

approved