A337870 The number of random walks on the simple square lattice that start at the origin (0,0) and pass through (1,0) after 2n+1 steps before having returned to the origin.

1, 2, 16, 166, 1934, 24076, 312906, 4191822, 57433950, 800740450, 11319707546, 161841539812, 2335765140994, 33979681977530, 497696233487200, 7332776490675630, 108595186409772174, 1615573668169487898, 24132221328987714066

Offset

0,2

Comments

The number of walks that take one of the four directions U, D, R, L which arrive at (1,0) is zero if the number of steps is even. For odd number of steps we count the walks that start at (0,0) pass through any set of points that are not {(0,0),(1,0)} and arrive at (1,0).

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

Links

Table of n, a(n) for n=0..18.

Maple

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

Crossrefs

Cf. A002894, A060150, A275912, A337869.

Sequence in context: A216598 A219397 A275912 * A181914 A089624 A217804

Adjacent sequences: A337867 A337868 A337869 * A337871 A337872 A337873

Keyword

nonn,walk

Author

R. J. Mathar, Sep 27 2020

Status

approved