%I #27 Sep 08 2022 08:46:18
%S 0,1,16576,10766601,1073790976,38147265625,705278736576,8308236966001,
%T 70368756760576,463255079498001,2500000075000000,11487432626662201,
%U 46221065046245376,166354152907593001,544488335559184576,1642102090850390625,4611686021648613376
%N Number of inequivalent 4 X 4 matrices with entries in {1,2,3,...,n} up to vertical and horizontal reflections.
%C Cycle index of dihedral group D2 acting on the 16 entries is (3s(2)^8 + s(1)^16)/4.
%H G. C. Greubel, <a href="/A283028/b283028.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_17">Index entries for linear recurrences with constant coefficients</a>, signature (17, -136, 680, -2380, 6188, -12376, 19448, -24310, 24310, -19448, 12376, -6188, 2380, -680, 136, -17, 1).
%F a(n) = n^8 * (n^8 + 3)/4.
%F From _Chai Wah Wu_, Dec 07 2018: (Start)
%F a(n) = 17*a(n-1) - 136*a(n-2) + 680*a(n-3) - 2380*a(n-4) + 6188*a(n-5) - 12376*a(n-6) + 19448*a(n-7) - 24310*a(n-8) + 24310*a(n-9) - 19448*a(n-10) + 12376*a(n-11) - 6188*a(n-12) + 2380*a(n-13) - 680*a(n-14) + 136*a(n-15) - 17*a(n-16) + a(n-17) for n > 16.
%F G.f.: -x*(x + 1)*(x^14 + 16558*x^13 + 10468387*x^12 + 882544028*x^11 + 20463263441*x^10 + 175065686258*x^9 + 626804969739*x^8 + 968894839176*x^7 + 626804969739*x^6 + 175065686258*x^5 + 20463263441*x^4 + 882544028*x^3 + 10468387*x^2 + 16558*x + 1)/(x - 1)^17. (End)
%F E.g.f.: (1/4)*x*exp(x)*(x^15 + 120*x^14 + 6020*x^13 + 165620*x^12 + 2757118*x^11 + 28936908*x^10 + 193754990*x^9 + 820784250*x^8 + 2141764056*x^7 + 3281882688*x^6 + 2734927356*x^5 + 1096193700*x^4 + 171804004*x^3 + 7144584*x^2 + 33148*x + 4). - _Stefano Spezia_, Dec 07 2018
%e For n=2 we get a(2)=16576 inequivalent 4 X 4 binary matrices up to vertical and horizontal reflections.
%p [n^8*(n^8+3)/4$n=0..18]; # _Muniru A Asiru_, Dec 07 2018
%t Table[n^8(n^8 + 3)/4,{n,0,30}]
%o (PARI) a(n) = n^8 * (n^8 + 3)/4; \\ _Altug Alkan_, Feb 27 2017
%o (Python) def A283028(n): return n**8*(n**8 + 3)/4 # _Indranil Ghosh_, Feb 27 2017
%o (Magma) [n^8*(n^8+3)/4: n in [0..20]]; // _G. C. Greubel_, Dec 07 2018
%o (Sage) [n^8*(n^8+3)/4 for n in range(20)] # _G. C. Greubel_, Dec 07 2018
%o (GAP) List([0..20], n -> n^8*(n^8+3)/4); # _G. C. Greubel_, Dec 07 2018
%Y Cf. A282612, A282613, A283026, A283027, A283029, A283030, A283031, A283033.
%Y Cf. A283032 (5 X 5 version), A282614 (3 X 3 version), A039623 (2 X 2 version).
%K nonn,easy
%O 0,3
%A _David Nacin_, Feb 27 2017