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A228583 The number of binary pattern classes in the (2,n)-rectangular grid with 8 '1's and (2n-8) '0's: two patterns are in same class if one of them can be obtained by a reflection or 180-degree rotation of the other.

%I A228583 #27 Sep 08 2022 08:46:05
%S A228583 0,0,0,0,1,15,135,777,3270,11034,31650,80190,184239,391105,777777,
%T A228583 1464255,2630940,4540836,7567380,12228780,19229805,29512035,44313643,
%U A228583 65239845,94345218,134229150,188145750,260129610,355138875,479217141,639675765,845298235,1106568312
%N A228583 The number of binary pattern classes in the (2,n)-rectangular grid with 8 '1's and (2n-8) '0's: two patterns are in same class if one of them can be obtained by a reflection or 180-degree rotation of the other.
%C A228583 Column 8 of A226048.
%H A228583 Vincenzo Librandi, <a href="/A228583/b228583.txt">Table of n, a(n) for n = 0..1000</a>
%H A228583 <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 A228583 a(n) = (1/4)*(binomial(2*n,8) + 3*binomial(n,4)).
%F A228583 a(n) = 9*a(n-1)-36*a(n-2)+84*a(n-3)-126*a(n-4)+126*a(n-5)-84*a(n-6)+36*a(n-7)-9*a(n-8)+a(n-9) n>8, with a(0)=0, a(1)=0, a(2)=0, a(3)=1, a(4)=15, a(5)=135, a(6)=777, a(7)=3270, a(8)=11034.
%F A228583 G.f.: -x^4*(3*x^4+18*x^3+36*x^2+6*x+1) / (x-1)^9. - _Colin Barker_, Sep 01 2013
%t A228583 CoefficientList[Series[- x^4 (3 x^4 + 18 x^3 + 36 x^2 + 6 x + 1) / (x - 1)^9, {x, 0, 50}], x] (* _Vincenzo Librandi_, Sep 04 2013 *)
%o A228583 (R) a <- 0
%o A228583     for(n in 1:40) a[n+1] <- (1/4)*(choose(2*n,8) + 3*choose(n,4))
%o A228583     a
%o A228583 (Magma) [(1/4)*(Binomial(2*n,8) + 3*Binomial(n,4)): n in [0..50]]; // _Vincenzo Librandi_, Sep 04 2013
%Y A228583 Cf. A226048.
%K A228583 nonn,easy
%O A228583 0,6
%A A228583 _Yosu Yurramendi_, _María Merino_, Aug 26 2013
%E A228583 More terms from _Vincenzo Librandi_, Sep 04 2013