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A211155 Number of 2 X 2 matrices having all terms in {-n,...,0,..,n} and odd determinant.

%I A211155 #19 Jul 16 2024 13:27:10
%S A211155 0,40,168,1056,2080,6120,9576,20608,28800,52200,68200,110880,138528,
%T A211155 208936,252840,360960,426496,583848,677160,896800,1024800,1321320,
%U A211155 1491688,1881216,2102400,2602600,2883816,3513888,3865120,4645800,5077800,6031360,6555648,7705896,8334760,9707040
%N A211155 Number of 2 X 2 matrices having all terms in {-n,...,0,..,n} and odd determinant.
%C A211155 A211154(n) + A211155(n) = (2n+1)^4.
%C A211155 For a guide to related sequences, see A210000.
%H A211155 Andrew Howroyd, <a href="/A211155/b211155.txt">Table of n, a(n) for n = 0..1000</a>
%H A211155 <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (1, 4, -4, -6, 6, 4, -4, -1, 1).
%F A211155 From _Chai Wah Wu_, Nov 27 2016: (Start)
%F A211155 a(n) = a(n-1) + 4*a(n-2) - 4*a(n-3) - 6*a(n-4) + 6*a(n-5) + 4*a(n-6) - 4*a(n-7) - a(n-8) + a(n-9) for n > 9.
%F A211155 G.f.: x*(-40*x^6 - 128*x^5 - 728*x^4 - 512*x^3 - 728*x^2 - 128*x - 40)/((x - 1)^5*(x + 1)^4). (End)
%p A211155 seq( 2*n*(1+n)*(1+3*n+3*n^2-(1+2*n)*(-1)^n), n=1..20); # _Mark van Hoeij_, May 13 2013
%t A211155 a = -n; b = n; z1 = 20;
%t A211155 t[n_] := t[n] = Flatten[Table[w*z - x*y, {w, a, b}, {x, a, b}, {y, a, b}, {z, a, b}]]
%t A211155 c[n_, k_] := c[n, k] = Count[t[n], k]
%t A211155 u[n_] := Sum[c[n, 2 k], {k, -2*n^2, 2*n^2}]
%t A211155 v[n_] := Sum[c[n, 2 k - 1], {k, -2*n^2, 2*n^2}]
%t A211155 Table[u[n], {n, 1, z1}] (* A211154 *)
%t A211155 Table[v[n], {n, 1, z1}] (* A211155 *)
%o A211155 (PARI) a(n)=2*n*(1+n)*(1+3*n+3*n^2-(1+2*n)*(-1)^n); \\ _Joerg Arndt_, May 14 2013
%Y A211155 Cf. A210000.
%K A211155 nonn
%O A211155 0,2
%A A211155 _Clark Kimberling_, Apr 05 2012
%E A211155 More terms from _Joerg Arndt_, May 14 2013
%E A211155 a(0)=0 prepended by _Andrew Howroyd_, May 05 2020