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A204243 Determinant of the n-th principal submatrix of A204242.

%I A204243 #30 Dec 10 2015 14:11:26
%S A204243 1,2,11,144,4149,251622,31340799,7913773980,4024015413705,
%T A204243 4106387069191890,8395359475529822355,34357677843892688699400,
%U A204243 281336437060919094044274525,4608419756389534634440592965950,150992374805715685629827976712244775
%N A204243 Determinant of the n-th principal submatrix of A204242.
%H A204243 Colin Barker, <a href="/A204243/b204243.txt">Table of n, a(n) for n = 1..80</a>
%H A204243 Wikipedia, <a href="https://en.wikipedia.org/wiki/Matrix_determinant_lemma">Matrix determinant lemma</a>
%F A204243 a(n) = (1 - Sum_{k=2..n} 1/(2^k-1)) * Product_{k=2..n} (2^k-1) = 2*A005329(n) - A203011(n). - _Robert Israel_, Nov 30 2015
%p A204243 f:= n -> (1 - add(1/(2^i-1),i=2..n))*mul(2^i-1,i=2..n):
%p A204243 seq(f(n),n=1..30); # _Robert Israel_, Nov 30 2015
%t A204243 f[i_, j_] := 0; f[1, j_] := 1; f[i_, 1] := 1; f[i_, i_] := 2^i - 1;
%t A204243 m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
%t A204243 TableForm[m[8]] (* 8x8 principal submatrix *)
%t A204243 Flatten[Table[f[i, n + 1 - i],
%t A204243   {n, 1, 12}, {i, 1, n}]]     (* A204242 *)
%t A204243 Table[Det[m[n]], {n, 1, 15}]  (* A204243 *)
%t A204243 Permanent[m_] :=
%t A204243   With[{a = Array[x, Length[m]]},
%t A204243    Coefficient[Times @@ (m.a), Times @@ a]];
%t A204243 Table[Permanent[m[n]], {n, 1, 15}]   (* A203011 *)
%o A204243 (PARI) vector(20, n, matdet(matrix(n, n, i, j, if(i==1, 1, if(j==1, 1, if(i==j, 2^i-1)))))) \\ _Colin Barker_, Nov 27 2015
%Y A204243 Cf. A005329, A203011, A204238, A204242.
%K A204243 nonn
%O A204243 1,2
%A A204243 _Clark Kimberling_, Jan 13 2012