%I
%S 1,2,16,166,1934,24076,312906,4191822,57433950,800740450,11319707546,
%T 161841539812,2335765140994,33979681977530,497696233487200,
%U 7332776490675630,108595186409772174,1615573668169487898,24132221328987714066
%N 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.
%C 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).
%C The ordinary generating function is a mix of inverses of sums and differences of the hypergeometric generating functions in A002894 and A060150. See Maple.
%p g002894 := hypergeom([1/2,1/2],[1],16*x^2) ;
%p g060150 := x*hypergeom([1,3/2,3/2],[2,2],16*x^2) ;
%p 1/2/(g002894g060150)1/2/(g002894+g060150) ;
%p taylor(%,x=0,40);
%p L := gfun[seriestolist](%) ; # includes zeros of even steps
%Y Cf. A002894, A060150, A275912, A337869.
%K nonn,walk
%O 0,2
%A _R. J. Mathar_, Sep 27 2020
