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A058001 Number of 3 X 3 matrices with entries mod n, up to row and column permutation.

%I A058001 #25 Nov 23 2024 13:21:03
%S A058001 1,36,738,8240,57675,289716,1144836,3780288,10865205,27969700,
%T A058001 65834406,143887536,295467263,575308020,1069960200,1911933696,
%U A058001 3298486761,5516122788,8972008810,14233690800,22078652211,33555443636,50058302988,73417387200,106006948125
%N A058001 Number of 3 X 3 matrices with entries mod n, up to row and column permutation.
%C A058001 Number of k X l matrices with entries mod n, up to row and column permutation is Z(S_k X S_l; n,n,...) where Z(S_k X S_l; x_1,x_2,...) is cycle index of Cartesian product of symmetric groups S_k and S_l of degree k and l, respectively.
%H A058001 Alois P. Heinz, <a href="/A058001/b058001.txt">Table of n, a(n) for n = 1..1000</a>
%H A058001 Marko R. Riedel, <a href="https://math.stackexchange.com/questions/2056708/">Number of equivalence classes of matrices</a>, Math Stackexchange.
%H A058001 Marko R. Riedel, <a href="/A058001/a058001.html.txt">Computing the cycle index for arbitary k x l matrices using Maple</a>
%H A058001 <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45,10,-1).
%F A058001 a(n) = (1/3!^2)*(n^9 + 6*n^6 + 9*n^5 + 8*n^3 + 12*n^2).
%F A058001 G.f.: x*(12*x^7+369*x^6+2514*x^5+4375*x^4+2360*x^3+423*x^2+26*x+1) / (x-1)^10. - _Colin Barker_, Jul 09 2013
%t A058001 CoefficientList[Series[x (12x^7+369x^6+2514x^5+4375x^4+2360x^3+423x^2+26x+1)/(x-1)^10,{x,0,30}],x] (* or *) LinearRecurrence[{10,-45,120,-210,252,-210,120,-45,10,-1},{0,1,36,738,8240,57675,289716,1144836,3780288,10865205},30] (* _Harvey P. Dale_, Nov 23 2024 *)
%Y A058001 Cf. A058002, A058003, A058004, A002724, A052271, A052272.
%Y A058001 Row n=3 of A246106.
%K A058001 easy,nonn
%O A058001 1,2
%A A058001 _Vladeta Jovovic_, Nov 04 2000