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

1, 2, 11, 144, 4149, 251622, 31340799, 7913773980, 4024015413705, 4106387069191890, 8395359475529822355, 34357677843892688699400, 281336437060919094044274525, 4608419756389534634440592965950, 150992374805715685629827976712244775

Offset: 1

Author

Clark Kimberling, Jan 13 2012

Keywords

• nonn

Links

• Colin Barker, Table of n, a(n) for n = 1..80
• Wikipedia, Matrix determinant lemma

Crossrefs

Cf. A005329, A203011, A204238, A204242.

Programs

Maple

f:= n -> (1 - add(1/(2^i-1),i=2..n))*mul(2^i-1,i=2..n):
seq(f(n),n=1..30); # Robert Israel, Nov 30 2015

Mathematica

f[i_, j_] := 0; f[1, j_] := 1; f[i_, 1] := 1; f[i_, i_] := 2^i - 1;
m[n_] := Table[f[i, j], {i, 1, n}, {j, 1, n}]
TableForm[m[8]] (* 8x8 principal submatrix *)
Flatten[Table[f[i, n + 1 - i],
  {n, 1, 12}, {i, 1, n}]]     (* A204242 *)
Table[Det[m[n]], {n, 1, 15}]  (* A204243 *)
Permanent[m_] :=
  With[{a = Array[x, Length[m]]},
   Coefficient[Times @@ (m.a), Times @@ a]];
Table[Permanent[m[n]], {n, 1, 15}]   (* A203011 *)

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

Formula

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