A337869 The number of random walks on the simple square lattice that return to the origin (0,0) after 2n steps and do not pass through (0,0) or (1,0) at intermediate steps.

3, 13, 106, 1073, 12142, 147090, 1865772, 24463905, 328887346, 4508608610, 62781858592, 885513974674, 12624162072740, 181611275997040, 2633023723495116, 38431604042148681, 564258290166041298, 8327627696761062714, 123471550301117915892

Offset: 1

Author

R. J. Mathar, Sep 27 2020

Keywords

• nonn
• walk

Comments

The number of walks on the simple square lattice that take one of the four directions U, D, R, L at each step and return to zero is zero if the number of steps is odd. If the number of steps is even, the sequence counts walks that start at (0,0), return to (0,0) and never pass through (0,0) or (1,0) in between.

The ordinary generating function is a mix of inverses of sums and differences of the hypergeometric generating functions in A002894 and A060150. See Maple.

Examples

			Example: a(1)=3 counts the walks UD, DU, LR (but not RL which would pass (1,0)) of 2 steps that return to the origin.
		

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, A060150, A337870.

Programs

Maple

g002894 := hypergeom([1/2,1/2],[1],16*x^2) ;
g060150 := x*hypergeom([1,3/2,3/2],[2,2],16*x^2) ;
1-1/2/(g002894+g060150)-1/2/(g002894-g060150) ;
taylor(%,x=0,40);
gfun[seriestolist](%) ; # includes zeros of odd steps