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

Author

R. J. Mathar, Sep 29 2020

Keywords

• nonn
• easy
• walk

Examples

			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.
		

Links

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

Crossrefs

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

Cf. A001791.

Programs

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

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