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%I #16 Dec 18 2020 11:50:30
%S 1,24,6384,2306904,964948464,439331916888,211383647188320,
%T 105734905550405400,54434276297806242480,28652982232251791825880,
%U 15350736081585866511795024,8343014042738696079671066904,4588687856038215036178166258304
%N 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).
%H Alois P. Heinz, <a href="/A181749/b181749.txt">Table of n, a(n) for n = 0..350</a>
%F Recursion: see Maple program. - _Alois P. Heinz_, Aug 31 2014
%F a(n) ~ 8 * 5^(4*n-1) / (3*sqrt(3) * (Pi*n)^(3/2)). - _Vaclav Kotesovec_, Sep 03 2014
%e a(1) = 24 because there are 24 rook paths from (0..0) to (1..1).
%p a:= proc(n) option remember; `if`(n<4, [1, 24, 6384, 2306904][n+1],
%p ((44148546*n^7-417566955*n^6+1582366209*n^5-3082719955*n^4
%p +3301523581*n^3-1923587242*n^2+559133416*n-61892160)*(n-1)^2*
%p a(n-1) -2*(n-2)*(131501097*n^8-1572004161*n^7+7935973542*n^6
%p -21971456652*n^5+36200366619*n^4-35926876063*n^3+20608609302*n^2
%p -6086148644*n+688049040)*a(n-2) +(393838614*n^7-4640973051*n^6
%p +22263043023*n^5-55659442951*n^4+77029268163*n^3
%p -57647348158*n^2+20864000120*n-2733950400)*(n-3)^2*a(n-3)
%p -5000*(34983*n^4-138138*n^3+184101*n^2-92498*n+14640)*(n-3)^2*
%p (n-4)^3*a(n-4))/ (2*n^3*(464360-1015046*n+808413*n^2
%p -278070*n^3+34983*n^4)*(n-1)^2))
%p end:
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Aug 31 2014
%t 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]}]];
%t a[n_] := b[Array[n&, 4]];
%t a /@ Range[0, 20] (* _Jean-François Alcover_, Dec 18 2020, after _Alois P. Heinz_ in A181731 *)
%Y Row d=4 of A181731.
%K nonn
%O 0,2
%A _Manuel Kauers_, Nov 16 2010
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