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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.

%I A337869 #8 Sep 28 2020 10:24:50
%S A337869 3,13,106,1073,12142,147090,1865772,24463905,328887346,4508608610,
%T A337869 62781858592,885513974674,12624162072740,181611275997040,
%U A337869 2633023723495116,38431604042148681,564258290166041298,8327627696761062714,123471550301117915892
%N 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.
%C A337869 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.
%C A337869 The ordinary generating function is a mix of inverses of sums and differences of the hypergeometric generating functions in A002894 and A060150. See Maple.
%H A337869 R. J. Mathar, <a href="/A337869/a337869.pdf">Random Walk on the Square Lattice: Return to (0,0) with or without passing (1,0)</a> (Sep 2020)
%e A337869 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.
%p A337869 g002894 := hypergeom([1/2,1/2],[1],16*x^2) ;
%p A337869 g060150 := x*hypergeom([1,3/2,3/2],[2,2],16*x^2) ;
%p A337869 1-1/2/(g002894+g060150)-1/2/(g002894-g060150) ;
%p A337869 taylor(%,x=0,40);
%p A337869 gfun[seriestolist](%) ; # includes zeros of odd steps
%Y A337869 Cf. A002894, A060150, A337870.
%K A337869 nonn,walk
%O A337869 1,1
%A A337869 _R. J. Mathar_, Sep 27 2020