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

0,2

Links

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

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

Example

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

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
      +3301523581*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
      +22263043023*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 *)

Crossrefs

Row d=4 of A181731.

Sequence in context: A163576 A145408 A175672 * A266312 A181232 A275054

Adjacent sequences: A181746 A181747 A181748 * A181750 A181751 A181752

Keyword

nonn

Author

Manuel Kauers, Nov 16 2010

Status

approved