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 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
 1, 2, 16, 166, 1934, 24076, 312906, 4191822, 57433950, 800740450, 11319707546, 161841539812, 2335765140994, 33979681977530, 497696233487200, 7332776490675630, 108595186409772174, 1615573668169487898, 24132221328987714066 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified June 5 02:01 EDT 2023. Contains 363130 sequences. (Running on oeis4.)