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
Links
- Andrew Howroyd, Table of n, a(n) for n = 3..200
Programs
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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 *) -
PARI
a(n) = 2*n*binomial(2*n,n-2) \\ Charles R Greathouse IV, May 21 2015
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)
Extensions
Terms a(22) and beyond from Andrew Howroyd, Nov 05 2019