A186648 Number of walks f length n on a square lattice ending with x > 0 and y > 0.

0, 0, 2, 6, 38, 130, 662, 2380, 11174, 41226, 185642, 695860, 3055670, 11576916, 49995220, 190876696, 814610854, 3128164186, 13233277634, 51046844836, 214488337418, 830382690556, 3470405605900, 13475470680616, 56073057254198, 218269673491780

Offset

0,3

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Formula

E.g.f.: ((e^(2*x)-I_0(2*x))/2)^2. - Benjamin Phillabaum, Mar 06 2011

From Benedict W. J. Irwin, May 25 2016: (Start)

If n is even, a(n) = 2^(n-2)*(2^n - 2*2F1((1-n)/2,-n/2;1;1) + n!*Gamma((n+1)/2))/(sqrt(Pi)*Gamma(1 + n/2)^3),

If n is odd, a(n) = 2^(n-2)*(2^n - 2*2F1((1-n)/2,-n/2;1;1)).

(End)

D-finite with recurrence n^2*(n-1)*(75*n-313)*a(n) -2*(334*n^2-1857*n+2052)*(n-1)^2*a(n-1) +8*(68*n^4-1240*n^3+6749*n^2-13464*n+9090)*a(n-2) +32*(300*n^4-2626*n^3+7387*n^2-6699*n-297)*a(n-3) -128*(218*n^2-1444*n+2429)*(-3+n)^2*a(n-4) +512*(2*n-9)*(17*n-105)*(-4+n)^2*a(n-5)=0. - R. J. Mathar, Feb 08 2021

Example

a(3) = 6 {UUR,URU,RUU,RRU,RUR,URR}. Note: you can also go Left or Down, however that appears at the fourth sequence which is too large to put in this space.

Mathematica

Table[(CoefficientList[Series[(1/2 (E^(2 x) - (BesselI[0, 2 x])))^2, {x, 0, len}], x] Range[0, len]!)[[n + 1]], {n, 0, 25}]

Crossrefs

Cf. A187151.

Sequence in context: A117266 A259437 A202737 * A372309 A013033 A233130

Adjacent sequences: A186645 A186646 A186647 * A186649 A186650 A186651

Keyword

nonn,walk

Author

Benjamin Phillabaum, Mar 06 2011

Status

approved