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%I #23 Oct 20 2016 23:45:05
%S 54,1140,26550,605556,14330736,340860960,8264889270,201651836100,
%T 4978317147804,123546256876224,3090501687886704,77632745316063840,
%U 1961313438507566400,49717549985405892480,1265749551338006549430,32312920048897640674500,827693426702217868006500
%N Number of paths in the lattice [0..n] X [0..n] X [0..n] which do not pass through the point (floor(n/2), floor(n/2), floor(n/2)). Number of paths through a lattice containing a "hole".
%H Robert Israel, <a href="/A071803/b071803.txt">Table of n, a(n) for n = 2..700</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LatticePath.html">Lattice Path.</a>
%F a(n) = s(3, n) - s(3, floor(n/2)) * s(3, ceiling(n/2)), where s(3,n) = A006480(n).
%F G.f.: 4F3(1/6,1/3,2/3,5/6; 1/2,1/2,1; 729*x^2) - 4F3(1/3,1/3,2/3,2/3; 1,1,1; 729*x^2) + 6*x*( 4F3(2/3,5/6,7/6,4/3; 1,3/2,3/2; 729*x^2) - 4F3(1/3,2/3,4/3,5/3; 1,2,2; 729*x^2)). - _Benedict W. J. Irwin_, Oct 20 2016
%p g:= hypergeom([1/6,1/3,2/3,5/6],[ 1/2,1/2,1], 729*x^2) - hypergeom([1/3,1/3,2/3,2/3],[ 1,1,1], 729*x^2) + 6*x*( hypergeom([2/3,5/6,7/6,4/3],[ 1,3/2,3/2], 729*x^2) - hypergeom([1/3,2/3,4/3,5/3],[ 1,2,2], 729*x^2)):
%p S:= series(g,x,101):
%p seq(coeff(S,x,j),j=2..100); # _Robert Israel_, Oct 20 2016
%t Table[Factorial[3n]/Factorial[n]^3 - Factorial[3Floor[n/2]]Factorial[3 Ceiling[n/2]]/Factorial[Floor[n/2]]^3/Factorial[Ceiling[n/2]]^3, {n, 2, 20}]
%t Rest[Rest[CoefficientList[Series[HypergeometricPFQ[{1/6, 1/3, 2/3, 5/6}, {1/2, 1/2, 1}, 729 x^2] - HypergeometricPFQ[{1/3, 1/3, 2/3, 2/3}, {1, 1, 1}, 729 x^2] + 6 x (HypergeometricPFQ[{2/3, 5/6, 7/6, 4/3}, {1, 3/2, 3/2}, 729 x^2] - HypergeometricPFQ[{1/3, 2/3, 4/3, 5/3}, {1, 2, 2}, 729 x^2]), {x, 0, 20}], x]]] (* _Benedict W. J. Irwin_, Oct 20 2016 *)
%o (PARI) A006480(n)=(3*n)!/n!^3
%o a(n) = A006480(n) - A006480(n\2)*A006480((n+1)\2) \\ _Charles R Greathouse IV_, Oct 20 2016
%Y Cf. A006480, A071800, A071801.
%K easy,nice,nonn
%O 2,1
%A _T. D. Noe_, Jun 06 2002
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