A181749 The number of paths of a chess rook in a 4D hypercube, from (0..0) to (n..n), where the rook may move in steps that are multiples of (1,0..0), (0,1,0..0), ..., (0..0,1).

1, 24, 6384, 2306904, 964948464, 439331916888, 211383647188320, 105734905550405400, 54434276297806242480, 28652982232251791825880, 15350736081585866511795024, 8343014042738696079671066904, 4588687856038215036178166258304

Offset: 0

Author

Manuel Kauers, Nov 16 2010

Keywords

• nonn

Examples

			a(1) = 24 because there are 24 rook paths from (0..0) to (1..1).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..350

Crossrefs

Row d=4 of A181731.

Programs

Maple

a:= proc(n) option remember; `if`(n<4, [1, 24, 6384, 2306904][n+1],
      ((44148546*n^7-417566955*n^6+1582366209*n^5-3082719955*n^4
      [phone]*n^3-1923587242*n^2+559133416*n-61892160)*(n-1)^2*
      a(n-1) -2*(n-2)*(131501097*n^8-1572004161*n^7+7935973542*n^6
      -21971456652*n^5+36200366619*n^4-35926876063*n^3+20608609302*n^2
      -6086148644*n+688049040)*a(n-2) +(393838614*n^7-4640973051*n^6
      [phone]*n^5-55659442951*n^4+77029268163*n^3
      -57647348158*n^2+20864000120*n-2733950400)*(n-3)^2*a(n-3)
      -5000*(34983*n^4-138138*n^3+184101*n^2-92498*n+14640)*(n-3)^2*
      (n-4)^3*a(n-4))/ (2*n^3*(464360-1015046*n+808413*n^2
      -278070*n^3+34983*n^4)*(n-1)^2))
    end:
seq(a(n), n=0..20); # Alois P. Heinz, Aug 31 2014

Mathematica

b[l_List] := b[l] = If[Union[l]~Complement~{0} == {}, 1, Sum[Sum[b[Sort[ ReplacePart[l, i -> l[[i]] - j]]], {j, 1, l[[i]]}], {i, 1, Length[l]}]];
a[n_] := b[Array[n&, 4]];
a /@ Range[0, 20] (* Jean-François Alcover, Dec 18 2020, after Alois P. Heinz in A181731 *)

Formula

Recursion: see Maple program. - Alois P. Heinz, Aug 31 2014

a(n) ~ 8 * 5^(4*n-1) / (3*sqrt(3) * (Pi*n)^(3/2)). - Vaclav Kotesovec, Sep 03 2014