A337869 -id:A337869 - cp's OEIS frontend

A337900 The number of walks of length 2n on the square lattice that start from the origin (0,0) and end at the vertex (2,0).

1, 16, 225, 3136, 44100, 627264, 9018009, 130873600, 1914762564, 28210561600, 418151049316, 6230734868736, 93271169290000, 1401915345465600, 21147754404155625, 320042195924198400, 4857445984927644900, 73916947787011560000, 1127482124965160372100

Offset: 1

R. J. Mathar, Sep 29 2020

			a(2) = 16 counts the walks RRRL, RRLR, RLRR, LRRR, RRUD, RRDU, RDRU, RURD, RUDR, RDUR, URRD, DRRU, URDR, DRUR, UDRR, DURR of length 4.

R. J. Mathar, Random Walk on the Square Lattice: Return to (0,0) with or without passing (1,0) (Sep 2020)

Cf. A002894 (at (0,0)), A060150 (at (1,0)), A135389 (at (1,1)), A337901 (at (3,0)), A337902 (at (2,1)).

Cf. A001791.

egf := BesselI(0, 2*x)*BesselI(2, 2*x): ser := series(egf, x, 40):
seq((2*n)!*coeff(ser, x, 2*n), n = 1..19);  # Peter Luschny, Dec 05 2024

a(n) = [A001791(n)]^2.
G.f.: x*4F3(3/2, 3/2, 2, 2; 1, 3, 3; 16*x).
D-finite with recurrence (n-1)^2*(n+1)^2*a(n) - 4*n^2*(2*n-1)^2*a(n-1) = 0.
a(n) = (2n)!*[x^(2n)] BesselI(0, 2x)*BesselI(2, 2x). - Peter Luschny, Dec 05 2024

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.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Sep 27 2020

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.

Cf. A002894, A060150, A275912, A337869.

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

cp's OEIS Frontend

A337900 The number of walks of length 2n on the square lattice that start from the origin (0,0) and end at the vertex (2,0).

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