A071803 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".

54, 1140, 26550, 605556, 14330736, 340860960, 8264889270, 201651836100, 4978317147804, 123546256876224, 3090501687886704, 77632745316063840, 1961313438507566400, 49717549985405892480, 1265749551338006549430, 32312920048897640674500, 827693426702217868006500

Offset: 2

Author

T. D. Noe, Jun 06 2002

Keywords

• easy
• nice
• nonn

Links

• Robert Israel, Table of n, a(n) for n = 2..700
• Eric Weisstein's World of Mathematics, Lattice Path.

Crossrefs

Cf. A006480, A071800, A071801.

Programs

Maple

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)):
S:= series(g,x,101):
seq(coeff(S,x,j),j=2..100); # Robert Israel, Oct 20 2016

Mathematica

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}]
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 *)

PARI

A006480(n)=(3*n)!/n!^3
a(n) = A006480(n) - A006480(n\2)*A006480((n+1)\2) \\ Charles R Greathouse IV, Oct 20 2016

Formula

a(n) = s(3, n) - s(3, floor(n/2)) * s(3, ceiling(n/2)), where s(3,n) = A006480(n).

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