A257888 Number of nonintersecting (or self-avoiding) rook paths of length 2n+2 joining opposite corners of an n X n grid.

4, 36, 224, 1200, 5940, 28028, 128128, 572832, 2519400, 10943240, 47070144, 200880160, 851809140, 3592795500, 15085939200, 63102895680, 263083395960, 1093683448440, 4535210472000, 18764563053600, 77485731403080, 319402222692696, 1314511549519104

Offset

3,1

Links

Andrew Howroyd, Table of n, a(n) for n = 3..200

Formula

a(n) = 2*(n-1)*binomial(2*n-2, n-3).

a(n) = A114593(n^2).

a(n) = 4*A002055(n+3). - Alois P. Heinz, May 21 2015

From Benedict W. J. Irwin, Jul 13 2016: (Start)

G.f.: 2*(s-1+x*(7-5*s+2*x*(2*s-6+x)))/(s^3x^2), where s=sqrt(1-4*x).

E.g.f: 2*E^(2*x)*x*(BesselI(1,2*x)+2*BesselI(2,2*x)+BesselI(3,2*x)).

(End)

Mathematica

a[n_] := 2 Sum[Sum[
   Binomial[j + k, k]*Binomial[2 n - k - j - 1, n - k + 1], {k,
    n}], {j, 0, n - 2}]
CoefficientList[Series[(2(-1+Sqrt[1-4x]+x(7-5Sqrt[1-4x] +2x(-6+2Sqrt[ 1-4x] +x))))/ ((1-4x)^(3/2)x^2), {x, 0, 20}], x] (* Benedict W. J. Irwin, Jul 13 2016 *)

Prog

(PARI) a(n) = 2*n*binomial(2*n, n-2) \\ Charles R Greathouse IV, May 21 2015

Crossrefs

Cf. A002055, A114593.

Sequence in context: A291643 A054773 A074434 * A197424 A306094 A018217

Adjacent sequences: A257885 A257886 A257887 * A257889 A257890 A257891

Keyword

nonn,walk

Author

Theodore M. Mishura, May 12 2015

Extensions

Terms a(22) and beyond from Andrew Howroyd, Nov 05 2019

Status

approved