%I #49 Sep 08 2022 08:46:04

%S 0,0,24,3024,43680,303600,1413720,5085024,15249024,39929760,94109400,

%T 203889840,412293024,787083024,1431033240,2495102400,4195023360,

%U 6831849024,10817040024,16702719120,25217757600,37310399280,54198168024,77425845024,108932342400

%N (n^2)*(n^2-1)*(n^2-2)*(n^2-3).

%C Number of n X n matrices using all elements of {1,2,3,4} exactly once with other entries zero. [clarified by _Debashish Sharma_, Oct 13 2014]

%H Vincenzo Librandi, <a href="/A217574/b217574.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).

%F G.f.: -24*x^2*(x+1)*(x^4+116*x^3+606*x^2+116*x+1)/(x-1)^9. [_Colin Barker_, Oct 11 2012]

%F a(n) = 24 * A189345(n). [_Joerg Arndt_, Oct 12 2012]

%F a(n) = n^2*(n^6 - 6*n^4 + 11*n^2 - 6). - _Jon Perry_, Nov 08 2014

%F a(0)=0, a(1)=0, a(2)=24, a(3)=3024, a(4)=43680, a(5)=303600, a(6)=1413720, a(7)=5085024, a(8)=15249024, a(n)=9*a(n-1)-36*a(n-2)+84*a(n-3)- 126*a(n-4)+ *a(n-5)-84*a(n-6)+36*a(n-7)-9*a(n-8)+a(n-9). - _Harvey P. Dale_, Mar 02 2015

%e For n=3, there are 3024 such matrices, e.g. ((123),(400),(000)) and ((030),(140),(002)).

%t Table[(n^2) (n^2 - 1) (n^2 - 2) (n^2 - 3), {n, 0, 30}] (* _T. D. Noe_, Oct 10 2012 *)

%t CoefficientList[Series[-24 x^2 (x + 1) (x^4 + 116 x^3 + 606*x^2 + 116*x + 1)/(x-1)^9, {x, 0, 40}], x] (* _Vincenzo Librandi_, Oct 25 2014 *)

%t Table[Times@@(n^2-Range[0,3]),{n,0,30}] (* or *) LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{0,0,24,3024,43680,303600,1413720,5085024,15249024},30] (* _Harvey P. Dale_, Mar 02 2015 *)

%o (Maxima) makelist((n^2)*(n^2-1)*(n^2-2)*(n^2-3), n, 0, 25); /* _Martin Ettl_, Oct 11 2012 */

%o (PARI) a(n)=(n^2)*(n^2-1)*(n^2-2)*(n^2-3); /* _Joerg Arndt_, Oct 12 2012 */

%o (Magma) [(n^2)*(n^2-1)*(n^2-2)*(n^2-3): n in [0..30]]; // _Vincenzo Librandi_, Oct 25 2014

%K nonn,easy

%O 0,3

%A _Debashish Sharma_, Oct 07 2012