A337900 The number of walks of length 2n on the square lattice that start from the origin (0,0) and end at the vertex (2,0).

1, 16, 225, 3136, 44100, 627264, 9018009, 130873600, 1914762564, 28210561600, 418151049316, 6230734868736, 93271169290000, 1401915345465600, 21147754404155625, 320042195924198400, 4857445984927644900, 73916947787011560000, 1127482124965160372100

Offset

1,2

Comments

Also, the number of walks within Z^2 starting and ending at (0,0) and consisting of 2n steps taken from {(-1, 0), (0, -1), (0, 1), (1, 0)} such that the walk does not stay inside the half plane N X Z. - Richard Ehrenborg, Jun 18 2026

Links

Table of n, a(n) for n=1..19.

Richard Ehrenborg, Gábor Hetyei and Margaret Readdy, Parking trees and the toric g-vector of nestohedra,arXiv:2511.04815 [math.CO], 2026.

Richard J. Mathar, Random Walk on the Square Lattice: Return to (0,0) with or without passing (1,0) (Sep 2020)

Formula

a(n) = A001791(n)^2.

G.f.: x*4F3(3/2, 3/2, 2, 2; 1, 3, 3; 16*x).

D-finite with recurrence (n-1)^2*(n+1)^2*a(n) - 4*n^2*(2*n-1)^2*a(n-1) = 0.

a(n) = (2n)!*[x^(2n)] BesselI(0, 2x)*BesselI(2, 2x). - Peter Luschny, Dec 05 2024

a(n) ~ 16^n / (Pi*n). - Amiram Eldar, Oct 16 2025

Example

a(2) = 16 counts the walks RRRL, RRLR, RLRR, LRRR, RRUD, RRDU, RDRU, RURD, RUDR, RDUR, URRD, DRRU, URDR, DRUR, UDRR, DURR of length 4.

Maple

egf := BesselI(0, 2*x)*BesselI(2, 2*x): ser := series(egf, x, 40):
seq((2*n)!*coeff(ser, x, 2*n), n = 1..19);  # Peter Luschny, Dec 05 2024

Mathematica

a[n_] := Binomial[2*n, n-1]^2; Array[a, 20, 0] (* Amiram Eldar, Oct 16 2025 *)

Crossrefs

Cf. A001791.

Cf. A002894 (at (0,0)), A060150 (at (1,0)), A135389 (at (1,1)), A337901 (at (3,0)), A337902 (at (2,1)).

Sequence in context: A051822 A017438 A098301 * A014897 A048445 A028340

Adjacent sequences: A337897 A337898 A337899 * A337901 A337902 A337903

Keyword

nonn,easy,walk,changed

Author

R. J. Mathar, Sep 29 2020

Status

approved