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%I #41 Nov 06 2019 01:12:43
%S 4,36,224,1200,5940,28028,128128,572832,2519400,10943240,47070144,
%T 200880160,851809140,3592795500,15085939200,63102895680,263083395960,
%U 1093683448440,4535210472000,18764563053600,77485731403080,319402222692696,1314511549519104
%N Number of nonintersecting (or self-avoiding) rook paths of length 2n+2 joining opposite corners of an n X n grid.
%H Andrew Howroyd, <a href="/A257888/b257888.txt">Table of n, a(n) for n = 3..200</a>
%F a(n) = 2*(n-1)*binomial(2*n-2, n-3).
%F a(n) = A114593(n^2).
%F a(n) = 4*A002055(n+3). - _Alois P. Heinz_, May 21 2015
%F From _Benedict W. J. Irwin_, Jul 13 2016: (Start)
%F G.f.: 2*(s-1+x*(7-5*s+2*x*(2*s-6+x)))/(s^3x^2), where s=sqrt(1-4*x).
%F E.g.f: 2*E^(2*x)*x*(BesselI(1,2*x)+2*BesselI(2,2*x)+BesselI(3,2*x)).
%F (End)
%t a[n_] := 2 Sum[Sum[
%t Binomial[j + k, k]*Binomial[2 n - k - j - 1, n - k + 1], {k,
%t n}], {j, 0, n - 2}]
%t 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 *)
%o (PARI) a(n) = 2*n*binomial(2*n,n-2) \\ _Charles R Greathouse IV_, May 21 2015
%Y Cf. A002055, A114593.
%K nonn,walk
%O 3,1
%A _Theodore M. Mishura_, May 12 2015
%E Terms a(22) and beyond from _Andrew Howroyd_, Nov 05 2019
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