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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). 2
1, 24, 6384, 2306904, 964948464, 439331916888, 211383647188320, 105734905550405400, 54434276297806242480, 28652982232251791825880, 15350736081585866511795024, 8343014042738696079671066904, 4588687856038215036178166258304 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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

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Last modified January 28 05:01 EST 2021. Contains 340490 sequences. (Running on oeis4.)