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%I #10 Feb 23 2017 22:23:05
%S 0,1,168,5346,67840,496875,2544696,10151428,33693696,97135605,
%T 250525000,590412966,1291500288,2653631071,5169160920,9616725000,
%U 17188519936,29659392873,49607301096,80696066410,128032800000,198613915731,301875282808,450363792396
%N Number of inequivalent 3 X 3 matrices with entries in {1,2,3,..,n} up to vertical and horizontal reflections.
%C Cycle index of symmetry group is (2*s(2)^3*s(1)^3 + s(2)^4*s(1) + s(1)^9)/4.
%H Colin Barker, <a href="/A282614/b282614.txt">Table of n, a(n) for n = 0..1000</a>
%H <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 a(n) = n^5*(n+1)*(n^3-n^2+n+1)/4.
%F G.f.: x*(1 + 158*x + 3711*x^2 + 21820*x^3 + 39095*x^4 + 22254*x^5 + 3577*x^6 + 104*x^7) / (1 - x)^10. - _Colin Barker_, Feb 23 2017
%e The number of 3 X 3 binary matrices up to vertical and horizontal reflections is 168.
%t Table[(2n+1+n^4)n^5/4, {n, 0, 24}]
%o (PARI) concat(0, Vec(x*(1 + 158*x + 3711*x^2 + 21820*x^3 + 39095*x^4 + 22254*x^5 + 3577*x^6 + 104*x^7) / (1 - x)^10 + O(x^30))) \\ _Colin Barker_, Feb 23 2017
%Y Cf. A282613, A282614, A217331, A168555. (For 2x2 version see A039623.)
%K nonn,easy
%O 0,3
%A _David Nacin_, Feb 19 2017
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