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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.
3
3, 13, 106, 1073, 12142, 147090, 1865772, 24463905, 328887346, 4508608610, 62781858592, 885513974674, 12624162072740, 181611275997040, 2633023723495116, 38431604042148681, 564258290166041298, 8327627696761062714, 123471550301117915892
OFFSET
1,1
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.
EXAMPLE
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.
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
CROSSREFS
KEYWORD
nonn,walk
AUTHOR
R. J. Mathar, Sep 27 2020
STATUS
approved