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

Author

Lucas A. Brown, Aug 24 2020

Keywords

• nonn
• easy

Comments

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

Links

• 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.

Crossrefs

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

Programs

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

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). (End)